Character triples and Shoda pairs
Gurmeet K. Bakshi and Gurleen Kaur∗†
Centre for Advanced Study in Mathematics,
Panjab University, Chandigarh 160014, India

arXiv:1702.00955v1 [] 3 Feb 2017

email: gkbakshi@pu.ac.in and gurleenkaur992gk@gmail.com

Abstract
In this paper, a construction of Shoda pairs using character triples is given
for a large class of monomial groups including abelian-by-supersolvable and
subnormally monomial groups. The computation of primitive central idempotents and the structure of simple components of the rational group algebra
for groups in this class are also discussed. The theory is illustrated with examples.

Keywords : rational group algebra, primitive central idempotents, simple components, Shoda pairs, strong Shoda pairs, character triples, monomial groups.
MSC2000 : 16S34, 16K20, 16S35

1

Introduction

Given a finite group G, Shoda ([5], Corollary 45.4) gave a criterion to determine
whether an induced monomial representation of G is irreducible or not. Olivieri,
del Rı́o and Simón [14] rephrased Shoda’s theorem as follows:
If χ is a linear character of a subgroup H of G with kernel K, then the
induced character χG is irreducible if, and only if, the following hold:
(i) K E H, H/K is cyclic;
(ii) if g ∈ G and [H, g] ∩ H ⊆ K, then g ∈ H.
A pair (H, K) of subgroups of G satisfying (i) and (ii) above is called a Shoda pair
of G. For K E H 6 G, define:
X
b := 1
h,
H
|H| h∈H
∗
†

Research supported by NBHM, India, is gratefully acknowledged
Corresponding author

1

ε(H, K) :=

(

b
K,
H=K;
Q b b
(K − L), otherwise,

where L runs over all the minimal normal subgroups of H containing K properly,
and

e(G, H, K) := the sum of all the distinct G-conjugates of ε(H, K).
An important feature ([14], Theorem 2.1) of a Shoda pair (H, K) of G is that there
is a rational number α, necessarily unique, such that αe(G, H, K) is a primitive
central idempotent of the rational group algebra QG, called the primitive central
idempotent of QG realized by the Shoda pair (H, K). We’ll denote this α by
α(G,H,K) . For monomial groups, all the primitive central idempotents of QG are
realized by Shoda pairs of G. For the Shoda pair (H, K) of G, the case when
e(G, H, K) is a primitive central idempotent of QG, is of special interest, thus,
leading to the following definition of a strong Shoda pair. A strong Shoda pair [14]
of G is a pair (H, K) of subgroups of G satisfying the following conditions:
(i) K E H E NG (K);
(ii) H/K is cyclic and a maximal abelian subgroup of NG (K)/K;
(iii) the distinct G-conjugates of ε(H, K) are mutually orthogonal.
In [14], it is proved that if (H, K) is a strong Shoda pair of G, then it is also a Shoda
pair of G and e(G, H, K) is a primitive central idempotent of QG. The groups G
such that all the primitive central idempotents of QG are realized by strong Shoda
pairs of G are termed as strongly monomial groups. Examples of such groups
include abelian-by-supersolvable groups ([14], Theorem 4.4). The main reason for
defining strong Shoda pairs in [14] was that the authors were able to provide a
description of the structure of the simple component QGe(G, H, K) of QG for a
strong Shoda pair (H, K) of G.
The work in [14] thus leads to the problem of computing Shoda pairs of a
given finite group G and to provide a description of the structure of the simple
components of QG corresponding to the primitive central idempotents realized by
them. The interest is in fact in providing a method to obtain a set S of Shoda
pairs of G such that the mapping (H, K) 7→ α(G,H,K) e(G, H, K) defines a bijection
from S to the set of all primitive central idempotents of QG realized by Shoda
pairs of G. Such a set S is called a complete and irredundant set of Shoda pairs
of G, and has recently been provided by the first author with Maheshwary [2] for
normally monomial groups. For the work in this direction, also see [1] and [3].
In this paper, we plan to study the problem for the class C of all finite groups G
such that all the subgroups and quotient groups of G satisfy the following property:
2

either it is abelian or it contains a non central abelian normal subgroup. The groups
in C are known to be monomial ([11], Lemma 24.2). However, we have noticed
that C is not contained in the class of strongly monomial groups. Huppert ([11],
Theorem 24.3) proved that C contains all the groups G for which there is a solvable
normal subgroup N of G such that all Sylow subgroups of N are abelian and G/N
is supersolvable. In particular, C contains abelian-by-supersolvable groups. In view
of an important criterion of subnormally monomial groups given in [8] and [9], we
have shown in section 2 that C also contains all subnormally monomial groups and,
in particular, normally monomial groups. Our aim is to extend the work to the
class C.
An important tool which has turned out to be useful is Isaacs’s notion of character triples together with Clifford’s correspondence theorem. Following Isaacs ([12],
p.186), we have defined N-linear character triples of G for a normal subgroup N
of G. In view of Clifford’s correspondence theorem ([12], Theorem 6.11), for each
N-linear character triple of G, we have defined its direct Clifford correspondents,
which is another set of N-linear character triples of G with useful properties proved
in Theorem 1 of section 3. With its help, we have given, in section 4, a construction of Shoda pairs of groups in C. For each normal subgroup N of G, we have
constructed a rooted directed tree GN , whose particular leaves correspond to Shoda
pairs of G, if G ∈ C (Theorem 2). We have also explored the condition for the
collection of Shoda pairs corresponding to these leaves of GN as N runs over all
the normal subgroups of G to be complete and irredundant. In section 5, we have
given a new character free expression of α(G,H,K), where (H, K) is a Shoda pair of
G corresponding to a leaf of GN . This expression is in terms of the directed path
from the root to the corresponding leaf and enables us to provide a necessary and
sufficient condition for e(G, H, K) to be a primitive central idempotent of QG. In
section 6, we generalize Proposition 3.4 of [14] and determine the structure of the
simple components of QG for G ∈ C. Finally, in section 7, we provide illustrative
examples.

2

The class C of monomial groups

Throughout this paper, G denotes a finite group. By H ≤ G, H G, H E G, we
mean, respectively, that H is a subgroup, proper subgroup, normal subgroup of
G. Denote by [G : H], the index of H in G. Also NG (H) denotes the normalizer
T
of H in G and coreG (H) = x∈G xHx−1 is the largest normal subgroup of G
contained in H. For x, y ∈ G, [x, y] = x−1 y −1 xy is the commutator of x and
3

y, and CenG (x) = {g ∈ G | gx = xg} is the centralizer of x in G. Denote by
Irr G, the set of all complex irreducible characters of G. For a character χ of G,
ker χ = {g ∈ G | χ(g) = χ(1)} and Q(χ) denotes the field obtained by adjoining
to Q the character values χ(g), g ∈ G. If ψ is a character of a subgroup H of G
and x ∈ G, then ψ x is the character of H x = x−1 Hx given by ψ x (g) = ψ(xgx−1 ),
g ∈ H x . Denote by ψ G , the character ψ induced to G. For a subgroup A of H, ψA
denotes the restriction of ψ to A.
Let C denote the class of all finite groups G such that all the subgroups and
quotient groups of G satisfy the following property: either it is abelian or it contains a non central abelian normal subgroup. It follows from Lemma 24.2 of [11]
that the groups in C are monomial. Recall that a finite group is monomial if every
complex irreducible character of the group is induced by a linear character of a
subgroup. In this section, we compare C with the following classes of groups:
Ab

: Abelian groups

Nil

: Nilpotent groups

Sup

: Supersolvable groups

A

: Solvable groups with all the Sylow subgroups abelian

nM

: Normally monomial groups, i.e., groups with all the complex
irreducible characters induced from linear characters of normal
subgroups

sM

: Subnormally monomial groups, i.e., groups with all the complex
irreducible characters induced from linear characters of subnormal
subgroups

stM

: Strongly monomial groups

X

: Solvable groups G satisfying the following condition : For all
primes p dividing the order of G and for all subgroups A of G,
O p (A), the unique smallest normal subgroup of A such that
A/O p (A) is a p-group, has no central p-factor

X - by -Y : Groups G such that there exist a normal subgroup N with N ∈ X
and G/N ∈ Y.
We prove the following:
Proposition 1 The following statements hold:
(i) Ab-by-Nil ⊆ Ab-by-Sup ⊆ A-by-Sup ⊆ C;
(ii) ( nM ∪ Ab-by-Nil ) ⊆ sM ⊆ X ⊆ C;
(iii) A-by-Sup * X;
4

(iv) X * A-by-Sup;
(v) C * stM.
Proof. (i) Clearly, Ab-by-Nil ⊆ Ab-by-Sup ⊆ A-by-Sup. From ([11], Theorem
24.3), it follows that A-by-Sup ⊆ C. This proves (i).
(ii) It is obvious that nM ⊆ sM. We now show that Ab-by-Nil ⊆ sM. Let G ∈
Ab-by-Nil. Let A be a normal abelian subgroup of G such that G/A is nilpotent.
Let χ ∈ Irr G. It is already known that χ is monomial. By ([11], Lemma 24.8),
there exists a subgroup H of G containing A such that χ is induced from a linear
character on H. As H/A is a subgroup of the nilpotent group G/A, it is subnormal
in G/A. Consequently, H is subnormal in G. This proves that G ∈ sM. Next, by
([9], Theorem 3.7), we have sM ⊆ X. We now show that X ⊆ C. By Lemma
2.6 of [8], X is closed under taking subgroups and factor groups. Thus to prove
that X ⊆ C, we only need to show that every non abelian group in X contains a
non central abelian normal subgroup. Let G ∈ X. If G is nilpotent, then clearly
it has the desired property. If G is not nilpotent, then Lemma 2.7 of [8] implies
that σ(G), the socle of G, is non central in G. Also, in view of ([13], Lemma 3.11,
Problem 2A.5), σ(G) is abelian, as G is solvable. Hence G has the desired property
and it follows that X ⊆ C.
(iii) Consider the group G generated by a, b, c, d with defining relations: a2 =
b3 = c3 = d3 = 1, a−1 ba = b−1 , a−1 ca = c−1 , a−1 da = d, b−1 cb = cd, b−1 db =
d, c−1 dc = d. It is easy to see that G is supersolvable and hence belongs to
A -by-Sup. We’ll show that G 6∈ X. Let CSF be the class of all chiefly subFrobenius groups, i.e., all finite solvable groups G for which CenG (kL) is subnormal
in G whenever kL is an element of a chief factor K/L of G. It is easy to see that
supersolvable groups are chiefly sub-Frobenius, and hence G ∈ CSF . Also it is
known ([9], Theorem 3.8) that sM=CSF ∩ X. Now if G ∈ X, then it follows that
G is subnormally monomial. It can be shown that G has an irreducible character
of degree 3, denoted by χ say. If G is subnormally monomial, then χ is induced
from a linear character of a subnormal subgroup H of G. Also, χ(1) = 3 implies
that [G : H] = 3, which yields that H is in fact normal in G. However, G does not
have any normal subgroup of index 3, a contradiction. This proves that G 6∈ X
and (iii) follows.
(iv) Consider G=SmallGroup(192, 1025) in GAP[7] library. It can be checked using
GAP that G is subnormally monomial and hence belongs to the class X but it does
not belong to A-by-Sup.
(v) Simple computations using Wedderga[4] reveal that SmallGroup(1000, 86) is
5

not strongly monomial. However, it belongs to C. This proves (v). 

3

N -linear character triples

Let G be a finite group. Let H ≤ G and (H, A, ϑ) a character triple, i.e., A E H
and ϑ ∈ Irr A is invariant in H. We call it to be N-linear character triple of G, if
ϑ is linear and ker ϑG = N. For N-linear character triple (H, A, ϑ) of G, denote
by Irr(H|ϑ), the set of all irreducible characters ψ of H which lie above ϑ, i.e.,
f
the restriction ψA of ψ to A has ϑ as a constituent. Let Irr(H|ϑ)
be its subset
f
consisting of those ψ which satisfy ker ψ G = N. Denote by Lin(H|ϑ),
the subset of
f
Irr(H|ϑ) consisting of linear characters. Further, for the character triple (H, A, ϑ),
we fix a normal subgroup A(H,A,ϑ) of H of maximal order containing ker ϑ such
that A(H,A,ϑ) / ker ϑ is abelian. Note that there may be several choices of such

A(H,A,ϑ) , however, we fix one such choice for a given triple (H, A, ϑ). Observe that
A(H,A,ϑ) / ker ϑ always contains the center of H/ ker ϑ. However, it can be seen that
if G ∈ C and H/ ker ϑ is non abelian, then A(H,A,ϑ) / ker ϑ properly contains the
center of H/ ker ϑ. We shall later use this observation without any mention.
Given N- linear character triple (H, A, ϑ) of G, we provide a construction of the
set Cl(H, A, ϑ) of another N-linear character triples of G required for the purpose
of constructing Shoda pairs of G.
Construction of Cl(H, A, ϑ)
Let Aut(C|ϑ) be the group of automorphisms of the field C of complex numbers
which keep Q(ϑ) fixed. For brevity, denote A(H,A,ϑ) by A. Consider the action of
f
Aut(C|ϑ) on Lin(A|ϑ)
by setting
f
σ ∈ Aut(C|ϑ), ϕ ∈ Lin(A|ϑ).

σ.ϕ = σ ◦ ϕ,

f
Also H acts on Lin(A|ϑ)
by

f
h ∈ H, ϕ ∈ Lin(A|ϑ).

h.ϕ = ϕh ,

f
Notice that the two actions on Lin(A|ϑ)
are compatible in the sense that

f
h ∈ H, σ ∈ Aut(C|ϑ), ϕ ∈ Lin(A|ϑ).

σ.(h.ϕ) = h.(σ.ϕ),

f
This consequently gives an action of Aut(C|ϑ) ×H on Lin(A|ϑ).
Under this double
f
action, denote by Lin(A|ϑ), a set of representatives of distinct orbits of Lin(A|ϑ).
If H 6= A, set

Cl(H, A, ϑ) = {(IH (ϕ), A, ϕ) | ϕ ∈ Lin(A|ϑ)},
6

where IH (ϕ) = {g ∈ G | ϕg = ϕ} is the inertia group of ϕ in H. For H =
A, define Cl(H, A, ϑ) to be an empty set. Note that all the character triples
in Cl(H, A, ϑ) are N-linear character triples of G and we call them the direct
Clifford correspondents (abbreviated d.c.c.) of (H, A, ϑ). The name ‘direct Clifford
f H (ϕ)|ϕ) are Clifford
correspondents’ refers to the fact that the characters in Irr(I
f
correspondents of the characters in Irr(H|ϑ)
in view of the following theorem:

Theorem 1 Let G ∈ C and N a normal subgroup of G. Let (H, A, ϑ) be Nlinear character triple of G with H 6= A and Cl(H, A, ϑ) be as defined above. Let
A = A(H,A,ϑ) . Then
(i) for any (IH (ϕ), A, ϕ) ∈ Cl(H, A, ϑ), the following hold:
(a) A A. Furthermore, A = H = IH (ϕ) holds, if, and only if, H/ ker ϑ is
abelian;
(b) IH (ϕ) E NH (ker ϕ);
f H (ϕ)|ϕ) to
(c) the induction ψ 7→ ψ H defines an injective map from Irr(I
f
Irr(H|ϑ).

f
f H (ϕ)|ϕ)
(ii) for each χ ∈ Irr(H|ϑ),
there exists (IH (ϕ), A, ϕ) ∈ Cl(H, A, ϑ), ψ ∈ Irr(I
and σ ∈ Aut(C) such that χ = σ ◦ ψ H .

f H (ϕ1 )|ϕ1),
(iii) if (IH (ϕ1 ), A, ϕ1), (IH (ϕ2 ), A, ϕ2 ) ∈ Cl(H, A, ϑ), ψ1 ∈ Irr(I
f H (ϕ2 )|ϕ2 ) and σ ∈ Aut(C) are such that ψ H = σ ◦ ψ H , then
ψ2 ∈ Irr(I
2
1
ϕ1 = ϕ2 (= ϕ say), and in this case ψ2 = σ ◦ ψ1x for some x ∈ NH (ker ϕ).

Proof. (i) We first show that
A.

A

(1)

If H/ ker ϑ is abelian, then clearly A = H and therefore the above equation holds
trivially, as A 6= H. If H/ ker ϑ is non abelian, then A/ ker ϑ properly contains
the centre of H/ ker ϑ. However, ϑ being invariant in H, it follows that A/ ker ϑ
is contained in the centre of H/ ker ϑ. Therefore, eqn (1) follows. This proves (a).
Next, consider α ∈ IH (ϕ) and β ∈ NH (ker ϕ). Then
β −1 αβ ∈ IH (ϕ)
if, and only if,
[β −1 αβ, x] = β −1 [α, βxβ −1]β ∈ ker ϕ for all x ∈ A.

(2)

However, if x ∈ A, then βxβ −1 ∈ A, as A E H. This gives [α, βxβ −1] ∈ ker ϕ,
as α ∈ IH (ϕ). Consequently, β being in NH (ker ϕ), eqn (2) follows. This proves
7

(b). In view of Clifford’s correspondence theorem ([12], Theorem 6.11), ψ 7→ ψ H
defines an injective map from Irr(IH (ϕ)|ϕ) to Irr(H|ϑ). It can be checked that,
f H (ϕ)|ϕ) is mapped to Irr(H|ϑ).
f
under this map, Irr(I
This finishes the proof of (c).
f
(ii) Consider χ ∈ Irr(H|ϑ).
Let λ be an irreducible constituent of χA . We claim
f
that λ ∈ Lin(A|ϑ).
As A E H and ϑ is invariant in H, by ([12], Theorem 6.2), ϑ is
the only irreducible constituent of χA . However, λ being an irreducible constituent

of χA , it follows that λA is a constituent of χA . Hence ϑ is an irreducible constituent
of λA , and therefore ker ϑA ≤ ker λ. But ker ϑA = coreA (ker ϑ) = ker ϑ, as ker ϑ
is normal in H. This gives ker ϑ ≤ ker λ. Consequently, A/ ker ϑ being abelian,
it follows that λ is linear and moreover λA = ϑ. We now show that ker λG = N.

As hχ, λH i, the inner product of χ with λH is non zero, and χ is irreducible, we
have ker λH ≤ ker χ. Hence coreG (ker λH ) ≤ coreG (ker χ), which gives ker λG ≤
ker χG = N. Also, ker ϑ ≤ ker λ implies that N = coreG (ker ϑ) ≤ coreG (ker λ) =
ker λG . Hence the claim follows. Now choose ϕ ∈ Lin(A|ϑ) which lies in the orbit
of λ. This gives σ ∈ Aut(C|ϑ) and x ∈ H such that λ = σ ◦ ϕx . As λ is an
irreducible constituent of χA , it follows that ϕx is an irreducible constituent of
the restriction of σ −1 ◦ χ to A. However, A being normal in H, from Clifford’s
theorem ([12], Theorem 6.2), it follows that ϕ is an irreducible constituent of
the restriction of σ −1 ◦ χ to A. Consequently, Clifford’s correspondence theorem
provides ψ ∈ Irr(IH (ϕ)|ϕ) such that σ −1 ◦ χ = ψ H . It is easy to check that this ψ
f H (ϕ)|ϕ). Hence (ii) follows.
belongs to Irr(I
f H (ϕ1 )|ϕ1 ), ψ2 ∈ Irr(I
f H (ϕ2 )|ϕ2 ) are such that
(iii) Suppose ψ1 ∈ Irr(I
ψ2H = σ ◦ ψ1H ,

(3)

where σ ∈ Aut(C). By restricting to A, it follows from ([12], Theorem 6.2) that
ϕ2 = σ ◦ ϕh1

(4)

for some h ∈ H. Therefore, ϕ1 and ϕ2 lie in the same orbit under the double action
and hence ϕ1 = ϕ2 = (ϕ say). In this case, from eqn (4),
ϕ = σ ◦ ϕh ,

(5)

which on comparing the kernels yields h ∈ NH (ker ϕ). Now using eqn (5) and the
f H (ϕ)|ϕ). Thus
fact that IH (ϕ) E NH (ker ϕ), it is easy to see that σ ◦ ψ1h ∈ Irr(I
f H (ϕ)|ϕ) and, in view of eqn (3), they are same
σ ◦ ψ1h and ψ2 both belong to Irr(I

when induced to H. Consequently, the injectivity of the induction map in part (i)
implies ψ2 = σ ◦ ψ1h . This proves (iii) and completes the proof. 
8

4

A construction of Shoda pairs

We begin by recalling some basic terminology in graph theory. A graph is a pair
G = (V, E), where V is a non-empty set whose elements are termed vertices of G
and E is a set of unordered pairs of vertices of G. Each element {u, v} ∈ E, where
u, v ∈ V, is called an edge and is said to join the vertices u and v. If e = {u, v} ∈ E,
then e is said to be incident with both u and v. Further, if v ∈ V, the degree of v
is the number of edges in E that are incident with v. A walk in the graph G is a
sequence of vertices and edges of the form : v1 {v1 , v2 } v2 · · · {vn−1 , vn } vn , where
each edge {vi , vi+1 } is incident with the vertices vi and vi+1 immediately preceding
and succeeding it. A walk is termed path if all the vertices are distinct and is
called a cycle if it begins and ends with the same vertex and all other vertices are
distinct. A connected graph is the one in which any two vertices are connected by
a path. A connected graph which contain no cycles is called a tree.
A directed graph is a pair G = (V, E), where V is a non-empty set whose
elements are termed vertices and E is a set of ordered pairs of vertices of V. In a
directed graph, an edge e = (u, v) is said to be incident out of u and incident into
v. The terminology of directed walk and directed path is same as that in graph
but now the edges are directed in the same direction. In a directed graph, the
number of edges incident out of v is called out-degree of v and the number of edges
incident into v is called in-degree of v, where v ∈ V. The vertices of in-degree 0
are called source and those of out-degree 0 are called sink. In a directed graph,
there is an obvious underlying undirected graph whose vertex set is same as that
of the directed graph and there is an edge {u, v} if either (u, v) or (v, u) is an edge
in the directed graph. A directed graph is called a directed tree if its underlying
undirected graph is a tree. The sink vertices of a directed tree are termed as its
leaves. A directed tree is called a rooted directed tree if it has a unique source.
The unique source of a rooted directed tree is called its root.
We now proceed with the construction of Shoda pairs. Let G ∈ C and let N
be a normal subgroup of G. Consider the directed graph G = (V, E) whose vertex
set V consist of all N-linear character triples of G and there is an edge (u, v) ∈ E
if v is a direct Clifford correspondent(d.c.c.) of u. Clearly (G, N, 1N ) ∈ V, where
1N is the character of N which takes constant value 1. Let VN be the set of those
vertices v ∈ V for which there is a directed path from (G, N, 1N ) to v. Let EN
be the set of ordered pairs (u, v) ∈ E with u, v ∈ VN . Then GN = (VN , EN ) is a
directed subgraph of G. Observe that any vertex (H, A, ϑ) of GN with H = A is a
sink vertex.

9

Theorem 2 Let G ∈ C and N the set of all normal subgroups of G.
(i) For N ∈ N , the following hold:
(a) GN is a rooted directed tree with (G, N, 1N ) as its root;
(b) the leaves of GN of the type (H, H, ϑ) correspond to Shoda pairs of G.
More precisely, if (H, H, ϑ) is a leaf of GN , then (H, ker ϑ) is a Shoda
pair of G.
(ii) If (H ′ , K ′ ) is any Shoda pair of G, then there is a leaf (H, H, ϑ) of GN , where
N = coreG (K ′ ), such that (H ′ , K ′ ) and (H, ker ϑ) realize the same primitive
central idempotent of QG.
To prove the theorem, we need some preparation.
Lemma 1 For each vertex v of GN , there is a unique directed path from (G, N, 1N )
to v.
′
′
′
Proof. Let v1 (v1 , v2 ) v2 · · · (vn−1 , vn ) vn and v1′ (v1′ , v2′ ) v2′ · · · (vm−1
, vm
) vm
be two
′
directed paths from v1 = v1′ = (G, N, 1N ) to vn = vm
= v = (H, A, ϑ). Assume

that m ≤ n. We claim that vi = vi′ for 1 ≤ i ≤ m. We’ll prove it by induction
on i. For i = 1, we already have v1 = v1′ = (G, N, 1N ). Assume that vi = vi′
for some i < m. Write vj = (Hj , Aj , ϑj ) and vj′ = (Hj′ , A′j , ϑ′j ) for 1 ≤ j ≤ m.
From the construction of d.c.c., we have Ai+1 = A(Hi ,Ai ,ϑi ) and A′i+1 = A(Hi′ ,A′i ,ϑ′i ) .
As (Hi , Ai , ϑi ) = (Hi′, A′i , ϑ′i ), it follows immediately that Ai+1 = A′i+1 . Now ϑi+1
(resp. ϑ′i+1 ) being the restriction of ϑ to Ai+1 (resp. A′i+1 ) yields that ϑi+1 = ϑ′i+1 .
′
′
Further, as Hi+1 = IHi (ϑi+1 ) and Hi+1
= IHi′ (ϑ′i+1 ), it follows that Hi+1 = Hi+1
.
This proves the claim, which as a consequence implies that both vm and vn are
equal to v. This is not possible if m < n as no two vertices in a path are same. 

Lemma 2 The following statements hold for GN :
(i) If (u1 , v1 ) and (u2 , v2 ) are edges of GN with v1 = v2 , then u1 = u2 ;
(ii) If v1 {v1 , v2 } v2 · · · {vn−1 , vn } vn is a path in the underlying undirected graph
of GN , then there is a unique j, 1 ≤ j ≤ n, with the following:
(a) (vi+1 , vi ) ∈ EN for 1 ≤ i < j;
(b) (vi , vi+1 ) ∈ EN for j ≤ i < n.
Proof. (i) is a consequence of Lemma 1 and (ii) follows immediately from (i). It
may be mentioned that in the lemma, (a) is empty if j = 1, and (b) is empty if
j = n. 
10

Lemma 3 The underlying undirected graph of GN is a tree.
Proof. Let GN = (VN , EN ) be the underlying undirected graph of GN = (VN , EN ).
Clearly GN is a connected graph. To show that GN is a tree, it is enough to
prove that GN is disconnected after removing an edge (see [6], Theorem 3-5). Let
e = {u, v} ∈ EN and let E′N = EN \ {e}. We need to show that G′N = (VN , E′N ) is
disconnected. If not, then there is a path
v1 {v1 , v2 } v2 · · · {vn−1 , vn } vn

(6)

in G′N , where v1 = u and vn = v. As {u, v} ∈ EN , either (u, v) or (v, u) belongs
to EN . Suppose (u, v) ∈ EN . As the path given in eqn (6) is also a path in GN ,
there is a j, 1 ≤ j ≤ n so that part (ii) of Lemma 2 holds. If j < n, then
vn = v is a d.c.c. of vn−1 , which by Lemma 2(i) implies vn−1 = u. Consequently
{u, v} = {vn−1 , vn } ∈ E′N , which is not so. If j = n, then vi is a d.c.c. of vi+1 for all
1 ≤ i < n. Hence if vi = (Hi , Ai , ϑi ), then from Theorem 1, An An−1 · · · A1 .
However (u, v) = (v1 , vn ) ∈ EN implies A1
An , a contradiction. Using similar
arguments, it follows that (v, u) ∈ EN is not possible. This proves the lemma. 
We are now ready to prove the theorem. Recall from ([15], Proposition 1.1)
P
P
−1
is a primitive
that if χ ∈ Irr G, then eQ (χ) = χ(1)
σ∈Gal(Q(χ)/Q)
g∈G σ(χ(g))g
|G|
central idempotent of QG, where Gal(Q(χ)/Q) is the Galois group of Q(χ) over
Q.

Proof of Theorem 2 (i) By Lemma 3, GN is a directed tree. If the in-degree of
(G, N, 1N ) is non zero then there is a N-linear character triple (H, A, ϑ) of G such
that (G, N, 1N ) is a d.c.c. of (H, A, ϑ). Hence by Theorem 1(i)(a) A N. However
(H, A, ϑ) being N-linear character triple of G, we have ker ϑG = N, which gives
N ≤ A, a contradiction. This proves that (G, N, 1N ) is a source of GN . Next if
(H, A, ϑ) is a vertex of GN different from (G, N, 1N ), then there is a directed path
from (G, N, 1N ) to (H, A, ϑ), which implies that the in-degree of (H, A, ϑ) is non
zero. Hence (H, A, ϑ) can’t be a source. This proves (a). To prove (b), consider a
leaf of GN of the type (H, H, ϑ). Let v1 (v1 , v2 ) v2 · · · (vn−1 , vn ) vn be the directed
path from v1 = (G, N, 1N ) to vn = (H, H, ϑ). Let vi = (Hi , Ai , ϑi ), 1 ≤ i ≤ n. As
(Hi+1 , Ai+1 , ϑi+1 ) is a d.c.c. of (Hi , Ai , ϑi ), from Theorem 1
f i |ϑi ) for all ψ ∈ Irr(H
f i+1 |ϑi+1 ).
ψ Hi ∈ Irr(H

(7)

f i |ϑi ) for 1 ≤ i ≤ n.
ϑHi ∈ Irr(H

(8)

f n |ϑn ) = {ϑn } = {ϑ}, as
This is true for all 1 ≤ i < n. Observe that Irr(H
Hn = An = H. The repeated application of eqn (7) when ψ is ϑ implies that

11

For i = 1, the above equation, in particular, gives ϑG ∈ Irr G. This proves (b).
(ii) Let (H ′ , K ′ ) be a Shoda pair of G and let N = coreG (K ′ ). Let ψ be a linear
character on H ′ with kernel K ′ . We claim that there is a leaf (H, H, ϑ) of GN
such that (H ′ , K ′ ) and (H, ker ϑ) realize the same primitive central idempotent
of QG. From the definition of the primitive central idempotent of QG realized
by a Shoda pair given in [14], it follows that the primitive central idempotent
of QG realized by (H ′ , K ′ ) is eQ (ψ G ) and that realized by (H, ker ϑ) is eQ (ϑG ).
Let χ = ψ G . By Shoda’s theorem, χ ∈ Irr G. Also, ker χ = N. If N = G,
then (H ′, K ′ ) is clearly (G, G). Also in this case, GN is just the vertex (G, G, 1G )
which corresponds to the Shoda pair (G, G) = (H ′ , K ′ ). Thus we may assume that
f
N 6= G. Denote χ by χ1 . As χ1 ∈ Irr(G|1
N ), by Theorem 1, there is a d.c.c.,
f
(IG (ϕ), A(G,N,1 ) , ϕ) of (G, N, 1N ), χ2 ∈ Irr(IG (ϕ)|ϕ) and σ1 ∈ Aut(C) such that
N

χG
2.

χ1 = σ1 ◦
Put (H1 , A1 , ϑ1 ) = (G, N, 1N ), (H2 , A2 , ϑ2 ) = (IG (ϕ), A(G,N,1N ) , ϕ).
If H2 = A2 , stop. If not, again by Theorem 1, there is a d.c.c. (H3 , A3 , ϑ3 ) of
f 3 |ϑ3 ), σ2 ∈ Aut(C) such that
(H2 , A2 , ϑ2 ), χ3 ∈ Irr(H
2
χ2 = σ2 ◦ χH
3 .

Moreover, if this case arises, then by Theorem 1(i)(a), N = A1

(9)

A2

A3 ≤ G.

Again if H3 = A3 stop, otherwise continue. This process of continuing must stop
after finite number of steps as at nth step, there is an ascending chain
N = A1

···

A2

An ≤ G.

Suppose the process stops at nth step. Then we have character triples (Hi , Ai , ϑi ),
f i |ϑi ) and σi ∈ Aut(C) such that
1 ≤ i ≤ n, with Hn = An , χi ∈ Irr(H
i
χi = σi ◦ χH
i+1 ,

1 ≤ i < n.

(10)

The above equation yields
χ = σ ◦ χG
n,

where σ = σ1 ◦ σ2 ◦ · · · ◦ σn .

f n |ϑn ) = {ϑn }, and hence χn = ϑn . This gives χ = σ◦θG .
As Hn = An , we have Irr(H
n

′
′
Consequently eQ (χ) = eQ (ϑG
n ) and hence (H , K ) and (Hn , ker ϑn ) realize the same
primitive central idempotent of QG. This proves the claim and completes the proof

of theorem. 
For N ∈ N , denote by LN the set of leaves of GN of type (H, H, ϑ). Let SN be
the set of Shoda pairs of G corresponding to the leaves in LN . We have shown in
S
Theorem 2 that if G ∈ C, then the mapping from N ∈N SN to the set of primitive
12

central idempotents of QG realized by the Shoda pairs of G is surjective. In other
S
words, N ∈N SN is a complete set of Shoda pairs of G. We now begin to investigate
whether this set of Shoda pairs of G is irredundant, i.e., this map is injective.
Two Shoda pairs (H1 , K1 ) and (H2 , K2 ) of G are said to be equivalent if they
realize the same primitive central idempotent of QG.
Lemma 4 If N and N ′ are distinct normal subgroups of G, then the Shoda pair
corresponding to a leaf in LN can not be equivalent to that in LN ′ .
Proof. Let (H, H, ϑ) ∈ LN and (H ′ , H ′, ϑ′ ) ∈ LN ′ . Suppose (H, ker ϑ) is equivalent to (H ′ , ker ϑ′ ). Then eQ (ϑG ) = eQ (ϑ′G ), which gives ϑ′G = σ ◦ ϑG for some
σ ∈ Aut(C). Consequently, ker ϑ′G = ker σ ◦ ϑG = ker ϑG . Hence, N = N ′ . 
We next examine if distinct leaves in LN , for a fixed normal subgroup N of G,
can correspond to equivalent Shoda pairs. For this purpose, we need to fix some
terminology. If (H, H, ϑ) ∈ LN and v1 (v1 , v2 ) v2 · · · (vn−1 , vn ) vn is the directed
path from v1 = (G, N, 1N ) to vn = (H, H, ϑ), then we call n to be the height of
(H, H, ϑ) and term vi as the ith node of (H, H, ϑ), 1 ≤ i ≤ n. It may be noted
f i |ϑi ) for all
that if vi = (Hi , Ai , ϑi ), then from eqn (8), it follows that ϑHi ∈ Irr(H

1 ≤ i ≤ n.

Definition Let (H, H, ϑ) ∈ LN be of height n with (Hi , Ai , ϑi ) as its ith node,
1 ≤ i ≤ n. We call (H, H, ϑ) to be good if the following holds for all 1 < i ≤ n:
given x ∈ NHi−1 (ker ϑi ), there exist σ ∈ Aut(C) such that (ϑHi )x = σ ◦ ϑHi .

Remark 1 It may be noted that, for any normal subgroup N of G, the leaves of
LN of height 2 are always good.
Lemma 5 If N ∈ N and (H, H, ϑ) ∈ LN is good, then its corresponding Shoda
pair can not be equivalent to that of other leaves in LN .
Proof. Let (H, H, ϑ) and (H ′, H ′ , ϑ′ ) ∈ LN be distinct and let (H, H, ϑ) be good.
Let the height of (H, H, ϑ) and (H ′ , H ′ , ϑ′ ) be n and n′ respectively. Let (Hi , Ai , ϑi )
be the ith node of (H, H, ϑ), 1 ≤ i ≤ n, and (Hj′ , A′j , ϑ′j ) the j th node of (H ′, H ′ , ϑ′ ),
1 ≤ j ≤ n′ . We have
f i |ϑi ), 1 ≤ i ≤ n
ϑHi ∈ Irr(H
(11)

and

′
f ′ |ϑ′ ), 1 ≤ j ≤ n′ .
ϑ′Hj ∈ Irr(H
j j

(12)

Let k be the least positive integer such that (Hk , Ak , ϑk ) 6= (Hk′ , A′k , ϑ′k ). Clearly,
k 6= 1. By the definition of k, (Hk , Ak , ϑk ) and (Hk′ , A′k , ϑ′k ) are distinct d.c.c.’s of
13

(Hk−1 , Ak−1 , ϑk−1 ). The construction of d.c.c. yields Ak = Ak′ . In view of eqns
(11), (12), it follows from Theorem 1(iii) that
ϑ′Hk−1 6= σ ◦ ϑHk−1

(13)

for any σ ∈ Aut(C). We now show a contradiction to eqn (13), if (H, ker ϑ) and
(H ′ , ker ϑ′ ) are equivalent. We have that Hi = Hi′ and ϑi = ϑ′i for 1 ≤ i ≤ k − 1.
f 2 |ϑ2 ).
Also H1 = H1′ = G. From eqns (11) and (12), ϑH2 and ϑ′H2 belong to Irr(H

If (H, ker ϑ) and (H ′ , ker ϑ′ ) are equivalent, then eQ (ϑ′G ) = eQ (ϑG ), which gives
ϑ′H1 = σ1 ◦ ϑH1 ,

for some σ1 ∈ Aut(C).

Hence, by Theorem 1(iii), there exist x ∈ NH1 (ker ϑ2 ) such that
ϑ′H2 = σ1 ◦ (ϑH2 )x .

(14)

Also, (H, H, ϑ) being good, there exists τ ∈ Aut(C) such that
(ϑH2 )x = τ ◦ ϑH2 .

(15)

Let σ2 = σ1 ◦ τ . From eqns (14) and (15),
ϑ′H2 = σ2 ◦ ϑH2 .
Now repeating this process with H1 , H2 replaced by H2 , H3 respectively, we obtain
that
ϑ′H3 = σ3 ◦ ϑH3 , for some σ3 ∈ Aut(C).
Continuing this process, we obtain after k − 1 steps that
ϑ′Hk−1 = σk−1 ◦ ϑHk−1 , for some σk−1 ∈ Aut(C).
This contradicts eqn (13) and completes the proof. 
Theorem 3 For G ∈ C,

S

N ∈N

SN is a complete irredundant set of Shoda pairs of

G if, and only if, the leaves in LN are good for all N ∈ N .
S
Proof. Suppose N ∈N SN is a complete irredundant set of Shoda pairs of G ∈ C.
Let N ∈ N . Let (H, H, ϑ) ∈ LN be of height n with (Hi , Ai , ϑi ) its ith node,
1 ≤ i ≤ n. We’ll show that (H, H, ϑ) is good. Let 2 ≤ i ≤ n and x ∈ NHi−1 (ker ϑi ).
f i |ϑi ). Proceeding as in the proof of part (ii) of
By Theorem 1(i)(c), (ϑHi )x ∈ Irr(H

Theorem 2, there exists (H ′ , H ′, ϑ′ ) ∈ LN such that

(ϑHi )x = σ ◦ ϑ′Hi , for some σ ∈ Aut(C).
14

(16)

S
Inducing to G, we get ϑG = σ ◦ϑ′G , which gives ϑ = ϑ′ and H = H ′ , as N ∈N SN is
complete and irredundant. Consequently, eqn (16) implies that (H, H, ϑ) is good.
This proves ‘only if’ statement. The ‘if’ statement follows from Theorem 2 and
Lemma 5. 
As a consequence, the above theorem yields the following result proved in ([2],
Theorem 1(i)):
S
Corollary 1 If G is a normally monomial group, then N ∈N SN is a complete
irredundant set of Shoda pairs of G.

Proof. It is enough to show that all the character triples in

S

N ∈N

LN are good.

Let (H, H, ϑ) ∈ LN , where N ∈ N . If N = G, then (H, H, ϑ) = (G, G, 1G ), which
G
f
is clearly good. Assume N 6= G. As ϑG ∈ Irr(G|1
N ), ker ϑ = N. Hence, by ([10],
Lemma 2.2), ϑG (1) = [G : A(G,N,1N ) ]. However, ϑG (1) = [G : H]. Therefore, we
have
[G : A(G,N,1N ) ] = [G : H]

(17)

Let the height of (H, H, ϑ) be n and let (Hi , Ai , ϑi ) be its ith node. As (H2 , A2 , ϑ2 )
is d.c.c. of (H1 , A1 , ϑ1 ) = (G, N, 1N ), we have A2 = A(G,N,1N ) . Also in view of
Theorem 1(i)(a), we have N = A1 A2 · · · An = H. This gives A(G,N,1N )
H, if n > 2, in which case eqn (17) can’t hold. Hence we must have n = 2, which
in view of remark 1, implies that (H, H, ϑ) is good. 
Remark 2 Later from remark 3, it will follow that if G is a normally monomial
S
group, then the Shoda pairs in N ∈N SN are strong Shoda pairs of G.

5

Idempotents from Shoda pairs

We continue to use the notation developed in the previous section. Given a group
G ∈ C, we have shown in the previous section that any Shoda pair of G is equivalent
S
to (H, ker ϑ), where (H, H, ϑ) is a character triple in N ∈N LN , and it realizes
the primitive central idempotent eQ (ϑG ) of QG. In this section, we examine the
expression of eQ (ϑG ) in terms of e(G, H, K), where K = ker ϑ. In [14], Olivieri,

del Rı́o and Simón proved that eQ (ϑG ) = α(G,H,K)e(G, H, K), where α(G,H,K) =
[CenG (ε(H,K)):H]
. The following theorem gives a new character free expression of
[Q(ϑ):Q(ϑG )]
α(G,H,K) and also provides a necessary and sufficient condition for α(G,H,K) = 1. It
may be mentioned that α(G,H,K) = 1 is a necessary condition for (H, K) to be a
strong Shoda pair of G.
Theorem 4 Let G ∈ C and N a normal subgroup of G. Let (H, H, ϑ) ∈ LN be of
height n with (Hi , Ai , ϑi ) as its ith node, 1 ≤ i ≤ n. Let K = ker ϑ. Then
15

(i) α(G,H,K) =

[CenG (ε(H,K)):CenHn−1 (ε(H,K))]
Q
;
2≤i≤n−1 [CenHi−1 (e(Hi ,H,K)):Hi ]

(ii) α(G,H,K) = 1 if, and only if, CenHi−1 (e(Hi , H, K)) = CenHi−1 (ε(H, K))Hi for
all 2 ≤ i ≤ n − 1;
(iii) if (H, H, ϑ) is good, then in the above statements CenHi−1 (e(Hi , H, K)) can
be replaced by NHi−1 (ker ϑi ) for all 2 ≤ i ≤ n − 1.
We first prove the following:
Lemma 6 Let G be a finite group and K E H ≤ G with H/K cyclic. Let A E H
and D = K ∩ A. Then
ε(A, D) = ε(H, K) + e
for some central idempotent e of QH orthogonal to ε(H, K).
Proof. Let ψ be a linear character on H with kernel K and let ϕ = ψA . By ([12],
Corollary 6.17), we have
X

ϕH =

hϕH , χψiχψ.

χ∈Irr H/A

As ϕ is invariant in H, we have (χψ)A = hϕ, (χψ)A iϕ = hϕH , χψiϕ, and hence
χψ(1) = hϕH , χψiϕ(1) = hϕH , χψi. Therefore,
ϕH =

X

(χψ)(1)χψ.

χ∈Irr H/A

P
Let n=[A:D] and
the collection of all the irreducible constituents of (ϕi )H ,
1 ≤ i ≤ n with (i, n) = 1. In other words,
X

= {χψ i | χ ∈ Irr H/A, 1 ≤ i ≤ n, (i, n) = 1}.

Consider the natural action of Aut(C) on

P

. Under this action, let χ1 , χ2 , · · · , χr

be the representatives of distinct orbits with χ1 = ψ. It can be checked that orb(χi ),
the orbit of χi , is given by {σ ◦ χi | σ ∈ Gal(Q(χi )/Q)}. Also, we have
X

1≤i≤n, (i,n)=1

(ϕi )H =

X

X

1≤i≤r σ∈Gal(Q(χi )/Q)

16

χi (1)σ ◦ χi .

Consequently,
X

1≤i≤r

X
1 X
χi (1)σ(χi (h))h−1
|H| h∈H
1≤i≤r
σ∈Gal(Q(χi )/Q)
X
X
1
=
(ϕi )H (h)h−1
|H| h∈H
1≤i≤n, (i,n)=1
X
1 X
(τ ◦ ϕ)H (h)h−1
=
|H| h∈H
τ ∈Gal(Q(ϕ)/Q)
X
1 X
(τ ◦ ϕ)(a)a−1
=
|A| a∈A

eQ (χi ) =

X

τ ∈Gal(Q(ϕ)/Q)

= ε(A, D).

As eQ (χ1 ) = eQ (ψ) = ε(H, K), the result follows. 
The following proposition is crucial in the proof of Theorem 4.
Proposition 2 Let G be a finite group. Let K E H 6 G with H/K cyclic and ψ
a linear character on H with kernel K. Suppose that there is a normal subgroup
A of G and a subgroup L of G such that A ≤ H ≤ L, ψA is invariant in L, and
L E NG (ker ψA ). Then the following hold:
(i) if ψ L ∈ Irr L, then the distinct G-conjugates of eQ (ψ L ) are mutually orthogonal;
(ii) if ψ G ∈ Irr G, then eQ (ψ G ) is the sum of all distinct G-conjugates of eQ (ψ L ).
Furthermore,
α(G,H,K) = α(L,H,K)

[CenG (ε(H, K)) : CenL (ε(H, K))]
.
[CenG (e(L, H, K)) : L]

Proof. (i) Denote ker ψA by D. First of all, we will show that
eQ (ψ L )eQ (ψ L )g = 0, if g 6∈ NG (D).

(18)

For this, it is enough to prove that
e(L, H, K)e(L, H, K)g = 0, if g 6∈ NG (D).

(19)

Let g 6∈ NG (D). We have
e(L, H, K)e(L, H, K)g = (

X

ε(H, K)x )(

x∈T

X

ε(H, K)y )g ,

y∈T

where T is a transversal of CenL (ε(H, K)) in L. Thus eqn (19) follows if we show
that for
ε(H, K)x ε(H, K)yg = 0,
17

for all x, y ∈ T.

By Lemma 6, ε(H, K)ε(A, D) = ε(H, K). This gives ε(H, K)x ε(A, D)x = ε(H, K)x
and ε(A, D)yg ε(H, K)yg = ε(H, K)yg . As ψA is invariant in L, we have D E L and
hence ε(A, D)x = ε(A, D) and ε(A, D)yg = ε(A, D)g . Thus, ε(H, K)x ε(H, K)yg
= ε(H, K)x ε(A, D)ε(A, D)g ε(H, K)yg = 0, as g 6∈ NG (D). This proves eqn (18),
which also yields
CenG (eQ (ψ L )) 6 NG (D).

(20)

Now, let g ∈ NG (D) \ CenG (eQ (ψ L )). Since ψ L ∈ Irr L and L E NG (D), it follows
that eQ (ψ L ) and eQ (ψ L )g are distinct primitive central idempotents of QL and
therefore eQ (ψ L )eQ (ψ L )g = 0. This proves (i).
(ii) Let T , T1 , T2 , T3 respectively be a right transversal of CenL (ε(H, K)) in G,
CenL (ε(H, K)) in L, L in CenG (eQ (ψ L )), CenG (eQ (ψ L )) in G. We have,
X
XX X
ε(H, K)x )yz
(
ε(H, K)g =
g∈T
z∈T
X2 x∈T1
X3 y∈T
e(L, H, K)yz
=
z∈T3 y∈T2

=

[CenG (eQ (ψ L )) : L] X
eQ (ψ L )z .
α(L,H,K)
z∈T
3

Also, it is easy to see that
X
ε(H, K)g = [CenG (ε(H, K)) : CenL (ε(H, K))]e(G, H, K)
g∈T

=

We know from part(i) that

[CenG (ε(H,K)):CenL (ε(H,K))]
eQ (ψ G ).
α(G,H,K)

X

eQ (ψ L )z is an idempotent. Also eQ (ψ G ) being an

z∈T3

idempotent, it follows immediately that
which gives the desired result. 

[CenG (eQ (ψL )):L]
α(L,H,K)

=

[CenG (ε(H,K)):CenL (ε(H,K))]
,
α(G,H,K)

The above proposition gives the following generalization of ([14], Corollary 3.6):

Corollary 2 Let (H, K) be a pair of subgroups of a finite group G and A be a
normal subgroup of G contained in H satisfying the following conditions:
(i) K E H E NG (D), where D = K ∩ A;
(ii) H/K is cyclic and a maximal abelian subgroup of NG (K)/K.
Then (H, K) is a strong Shoda pair of G and e(G, H, K) is a primitive central
idempotent of QG.
18

Proof. In view of (i), H = L satisfies the hypothesis of the above proposition.
Therefore, the above proposition yields that the distinct G-conjugates of ε(H, K)
are mutually orthogonal and hence (H, K) is a strong Shoda pair of G. 
We also have the following:
Corollary 3 Let G ∈ C, N a normal subgroup of G. Let (H, H, ϑ) ∈ LN . If
H E NG (ker ϑ2 ), then (H, ker ϑ) is a strong Shoda pair of G, where (H2 , A2 , ϑ2 ) is
the second node of (H, H, ϑ).
Proof. Let K = ker ϑ. From Theorem 2, (H, K) is a Shoda pair of G. By
considering A = A2 , L = H and ψ = ϑ in the above proposition, it follows that the
distinct G-conjugates of ε(H, K) are mutually orthogonal. Also ϑA2 =ϑ2 implies
that K ∩ A2 = ker ϑ2 and hence NG (K) 6 NG (ker ϑ2 ). Now H being normal in
NG (ker ϑ2 ), it follows that H E NG (K). Consequently, by ([14], Proposition 3.3),
it follows that (H, K) is a strong Shoda pair of G. 
Remark 3 From the above corollary, it follows immediately that if G ∈ C and
S
(H, H, ϑ) ∈ N ∈N LN is of height 2, then (H, ker ϑ) is a strong Shoda pair of G.

Proof of Theorem 4 (i) By taking L = A = Hn and G = Hn−1 in Proposition 2,
it follows that
α(Hn−1 ,H,K) = 1.
Also for 2 ≤ i ≤ n − 1, by taking L = Hi , G = Hi−1 , A = Ai , we have
α(Hi−1 ,H,K) = α(Hi ,H,K)

[CenHi−1 (ε(H, K)) : CenHi (ε(H, K))]
.
[CenHi−1 (e(Hi , H, K)) : Hi ]

Consequently
α(G,H,K) =
=

Q

[CenHi−1 (ε(H,K)):CenHi (ε(H,K))]
2≤i≤n−1
[CenHi−1 (e(Hi ,H,K)):Hi ]
[CenG (ε(H,K)):CenHn−1 (ε(H,K))]
Q
,
2≤i≤n−1 [CenHi−1 (e(Hi ,H,K)):Hi ]

as desired.
(ii) In view of (i), α(G,H,K) = 1 if, and only if,
Y

[CenHi−1 (ε(H, K)) : CenHi (ε(H, K))] =

2≤i≤n−1

Y

[CenHi−1 (e(Hi , H, K)) : Hi ].

2≤i≤n−1

But CenHi−1 (ε(H, K))/ CenHi (ε(H, K)) being isomorphic to a subgroup of
CenHi−1 (e(Hi , H, K))/Hi , the above equation holds if, and only if,
[CenHi−1 (ε(H, K)) : CenHi (ε(H, K))] = [CenHi−1 (e(Hi , H, K)) : Hi ],
19

for all i, 2 ≤ i ≤ n − 1, which yields the required result.
(iii) Let 2 ≤ i ≤ n − 1. Clearly x ∈ CenHi−1 (e(Hi , H, K)) if, and only if, x ∈
CenHi−1 (eQ (ϑHi )) if, and only if, eQ ((ϑHi )x ) = eQ (ϑHi ) if, and only if, (ϑHi )x =
σ ◦ ϑHi for some σ ∈ Aut(C). However, the later holds if, and only if, x ∈
NHi−1 (ker ϑi ), provided (H, H, ϑ) is good. This proves (iii). 

6

Simple components

Given G ∈ C, it follows from Theorem 2 that any simple component of QG is given
by QGeQ (ϑG ), where (H, H, ϑ) ∈ ∪N ∈N LN . Let’s now compute the structure of
QGeQ (ϑG ).
For a ring R, let U(R) be the unit group of R and Mn (R) the ring of n × n
matrices over R. Denote by R ∗στ G, the crossed product of the group G over the
ring R with action σ and twisting τ .
We begin with the following generalization of Proposition 3.4 of [14]:
Proposition 3 Let G be a finite group and (H, K) a Shoda pair of G. Let ψ, A, L
be as in Proposition 2. Then
QGeQ (ψ G ) ∼
= Mn (QLeQ (ψ L ) ∗στ CenG (eQ (ψ L ))/L),
where n = [G : CenG (eQ (ψ L ))], the action σ :CenG (eQ (ψ L ))/L → Aut(QLeQ (ψ L ))
maps x to the conjugation automorphism induced by x and the twisting
τ :CenG (eQ (ψ L ))/L × CenG (eQ (ψ L ))/L → U(QLeQ (ψ L )) is given by (g1 , g2 ) 7→ g,
where g ∈ L is such that g1 · g2 = g · g1 g2 for g1 , g2 ∈ CenG (eQ (ψ L ))/L.
Proof. Clearly QGeQ (ψ G ) is isomorphic to the ring EndQG (QGeQ (ψ G )) of QG
endomorphisms of QGeQ (ψ G ). As eQ (ψ G ) is the sum of distinct G-conjugates of
eQ (ψ L ), we have
EndQG (QGeQ (ψ G )) ∼
= Mn (EndQG (QGeQ (ψ L ))),
where n = [G : CenG (eQ (ψ L ))]. Also the map f 7→ f (eQ (ψ L )) yields isomorphism
EndQG (QGeQ (ψ L )) ∼
= eQ (ψ L )QGeQ (ψ L ).
But distinct G-conjugates of eQ (ψ L ) being orthogonal, we have eQ (ψ L )QGeQ (ψ L ) ∼
=
eQ (ψ L )QCeQ (ψ L ) = QCeQ (ψ L ), where C = CenG (eQ (ψ L )). Consequently, QGeQ (ψ G )
is isomorphic to Mn (QCeQ (ψ L )). Since L E NG (ker ψA ), eqn (20) gives L E C and
hence
QCeQ (ψ L ) ∼
= QLeQ (ψ L ) ∗σ C/L,
τ

20

where σ and τ are as in the statement. This completes the proof. 
We now proceed to compute the structure of the simple component QGeQ (ϑG )
of QG, where G ∈ C and (H, H, ϑ) ∈ ∪N ∈N LN .
Suppose that (H, H, ϑ) ∈ ∪N ∈N LN is of height n and (Hi , Ai , ϑi ) is its ith node,
1 ≤ i ≤ n. Let K = ker ϑ. Let 1 ≤ i ≤ n − 1. We must notice that G = Hi , L =
Hi+1 , A = Ai+1 and ψ = ϑ satisfy the hypothesis of Proposition 3. Let ki = [Hi :
CenHi (eQ (ϑHi+1 ))], σi the action of CenHi (eQ (ϑHi+1 ))/Hi+1 on QHi+1 eQ (ϑHi+1 ) by
conjugation, τi : CenHi (eQ (ϑHi+1 ))/Hi+1 ×CenHi (eQ (ϑHi+1 ))/Hi+1 → U(QHi+1 eQ (ϑHi+1 ))
the twisting given by (g1 , g2 ) 7→ g, where g ∈ Hi+1 is such that g1 · g2 = g · g1 g2 , for
g1 , g2 ∈ CenHi (eQ (ϑHi+1 ))/Hi+1 . Observe that CenHi (eQ (ϑHi+1 ))=CenHi (e(Hi+1 , H, K))
for all i.
Apply Proposition 3 with G = Hn−1 and A = L = Hn . It follows that
n−1
NHn−1 (K)/H),
QHn−1 eQ (ϑHn−1 ) ∼
= Mkn−1 (QHε(H, K) ∗στn−1

(21)

as eQ (ϑHn ) = ε(H, K) and CenHn−1 (ε(H, K)) = NHn−1 (K). Denote by R the
matrix ring on right hand side of the above isomorphism.
The action of CenHn−2 (e(Hn−1 , H, K))/Hn−1 on QHn−1 eQ (ϑHn−1 ) given by σn−2
induces its natural action on R given by x 7→ η ◦ σn−2 (x) ◦ η −1 , where x ∈
CenHn−2 (e(Hn−1 , H, K))/Hn−1 and η is the isomorphism in eqn (21). For notational convenience, we denote this action on R again by σn−2 . Similarly, the twisting η◦τn−2 from CenHn−2 (e(Hn−1 , H, K))/Hn−1 ×CenHn−2 (e(Hn−1 , H, K))/Hn−1 to
U(R) will again be denoted by τn−2 for convenience. Applying Proposition 3 with
G = Hn−2 , L = Hn−1 , A = An−1 , it follows that QHn−2 eQ (ϑHn−2 ) is isomorphic to
n−1
n−2
Mkn−2 ((Mkn−1 (QHε(H, K)∗στn−1
NHn−1 (K)/H)∗στn−2
CenHn−2 (e(Hn−1 , H, K)/Hn−1).

Continue this process, after n − 1 steps, we obtain the following:
Theorem 5 Let G ∈ C and (H, H, ϑ) ∈ ∪N ∈N LN . If (H, H, ϑ) is of height n and
(Hi , Ai , ϑi ) is its ith node, 1 ≤ i ≤ n, then QGeQ (ψ G ) is isomorphic to
n−1
n−2
Mk1 (Mk2 · · · (Mkn−1 (Q(ξk )∗στn−1
NHn−1 (K)/H)∗στn−2
· · ·∗στ11 CenH1 (e(H2 , H, K))/H2),

where k=[H : K], ξk is a primitive k th root of unity, and σi , τi , ki are as defined
above.

7

Examples

In this section, we illustrate the construction of Shoda pairs. We begin by observing
the following facts for an arbitrary group G in C:
21

1. The directed tree GN , when N = G, is just the vertex (G, G, 1G ), as Cl(G, G, 1G )
is an empty set. In this case, GN corresponds to the Shoda pair (G, G).
2. If N is a normal subgroup of G such that A(G,N,1N ) /N is cyclic, then
(G, N, 1N ) has only one d.c.c., namely, (IG (ϕ), A(G,N,1N ) , ϕ),
where ϕ can be taken to be any linear character on A(G,N,1N ) with kernel
f (G,N,1 ) |1N ). We have ϕN = 1N and
N. To show this, consider ϕ ∈ Lin(A
N

ker ϕG = N. The condition ϕN = 1N gives N ≤ ker ϕ ≤ A(G,N,1N ) . This

yields that ker ϕ is a normal subgroup of G, as A(G,N,1N ) /N is a cyclic normal
subgroup of G/N. Consequently, we have ker ϕ = ker ϕG = N, and thus ϕ
is a linear character on A(G,N,1N ) with kernel N. Conversely, it is clear that
f (G,N,1 ) |1N ).
any linear character on A(G,N,1N ) with kernel N belongs to Lin(A
N
f
Hence Lin(A(G,N,1N ) |1N ) consists precisely of all linear characters on A(G,N,1N )
with kernel N. It is easy to see that all these characters lie in the same orbit
under the double action. Consequently, there is only one d.c.c. of (G, N, 1N ),
as desired.
Further, if A(G,N,1N ) is such that A(G,N,1N ) /N is maximal among all the
abelian subgroups of G/N, then we have IG (ϕ) = A(G,N,1N ) and hence there
is no further d.c.c. of (IG (ϕ), A(G,N,1N ) , ϕ) and the process stops here and
the corresponding directed tree is as follows:
(A(G,N,1N ) , A(G,N,1N ) , ϕ)

(G, N, 1N )

Figure 1: GN
If A(G,N,1N ) /N is not maximal among all the abelian subgroups of G/N, then
IG (ϕ) 6= A(G,N,1N ) . In this case, we further need to compute the d.c.c. of
(IG (ϕ), A(G,N,1N ) , ϕ) in order to determine GN .
3. From the above fact, it follows that if N is a normal subgroup of G with
G/N cyclic, then its corresponding directed tree is as follows:

22

(G, G, ϕ)

(G, N, 1N )

Figure 2: GN
where ϕ is any linear character of G with ker ϕ = N and it yields the Shoda
pair (G, N) of G.
4. If N is a normal subgroup of G with G/N abelian but not cyclic then there
f
is no d.c.c. of (G, N, 1N ). As, in this case A(G,N,1 ) = G and ϕ ∈ Lin(G|1
N)
N

implies that ϕN = 1N and ker ϕG = N gives ker ϕ = N. Consequently, G/N

is cyclic, which is not the case. Hence the directed tree GN is just the vertex
(G, N, 1N ), which does not yield any Shoda pair as there is no leaf of the
required type.

5. Given N-linear character triple (H, A, ϑ) of G, consider the set K of all normal
subgroups K of A(H,A,ϑ) satisfying the following:
(i) A(H,A,ϑ) /K is cyclic;
(ii) K ∩ A = ker ϑ;
(iii) coreG (K) = N.
Let K1 , K2 , · · · , Kr be a set of representatives of K under the equivalence
relation defined by conjugacy of subgroups in H. If ϕi is a linear character
on A(H,A,ϑ) with kernel Ki , 1 ≤ i ≤ r, then
Cl(H, A, ϑ) = {(IH (ϕi ), A(H,A,ϑ), ϕi ) | 1 ≤ i ≤ r}.
f (H,A,ϑ) |ϑ). Let K = ker ϕ. Clearly A(H,A,ϑ)/K
To show this, consider ϕ ∈ Lin(A
is cyclic. Also ϕA = ϑ and ker ϕG = N implies that K ∩ A = ker ϑ
and coreG (K) = N. Hence K ∈ K. Thus we have shown that if ϕ ∈
f (H,A,ϑ) |ϑ), then ker ϕ ∈ K. Conversely, it is clear that if K ∈ K, then
Lin(A
f (H,A,ϑ) |ϑ). Furany linear character on A(H,A,ϑ) with kernel K lie in Lin(A
f (H,A,ϑ) |ϑ) lie in the same orbit under
thermore, note that ϕ1 and ϕ2 ∈ Lin(A
the double action if, and only if, ker ϕ1 and ker ϕ2 are conjugate in H. This
yields the desired result.

6. In the above fact, if (H, A, ϑ) is such that H/ ker ϑ is abelian, then it may be
noted that K, K ′ ∈ K are conjugate in H if, and only if, K = K ′ .
23

7.1

Example 1

Let G be the group generated by xi , 1 ≤ i ≤ 6, with the following defining
relations:
2 −1
5
2
5
5
x21 x−1
2 =x2 x3 =x4 =x3 =x5 =x6 =1,

[x2 , x1 ]=[x3 , x1 ]=[x3 , x2 ]=[x6 , x3 ]=[x6 , x4 ]=[x6 , x5 ]=1,
[x5 , x4 ]=x6 , [x5 , x1 ]=x4 x5 ,
[x6 , x1 ]=x26 , [x4 , x2 ]=x4 x26 , [x6 , x2 ]=x36 ,
[x5 , x2 ]=x5 x26 , [x5 , x3 ]=x35 x26 , [x4 , x3 ]=x34 x26 , [x4 , x1 ]=x24 x35 x46 .
This group is SmallGroup(1000,86) in GAP library. We have already mentioned
in section 2 that it belongs to C but it is not strongly monomial. There are 6
normal subgroups of G given by N1 =G, N2 =hx2 , x3 , x4 , x5 , x6 i, N3 =hx3 , x4 , x5 , x6 i,
N4 =hx4 , x5 , x6 i, N5 =hx6 i, N6 =h1i. We will compute the directed tree GNi , for all
i. From fact 1, GN1 is just the vertex (G, G, 1G ) and it corresponds to the Shoda
pair (G, G). For 2 ≤ i ≤ 4, G/Ni is cyclic. Therefore if ϕ1 , ϕ2 , ϕ3 are linear
characters on G with kernel N2 , N3 , N4 respectively, then, by fact 3 above, we
have the following:
(G, G, ϕ1 )

(G, G, ϕ2 )

(G, G, ϕ3 )

(G, N2 , 1N2 )

(G, N3 , 1N3 )

(G, N4 , 1N4 )

Figure 3: GN2

Figure 4: GN3

Figure 5: GN4

and the leaves (G, G, ϕ1 ), (G, G, ϕ2) and (G, G, ϕ3 ) yield the Shoda pairs (G, N2 ),
(G, N3 ), (G, N4 ) of G respectively.
Construction of GN5 : We first need to compute d.c.c. of (G, N5 , 1N5 ). Observe
that N4 /N5 is an abelian normal subgroup of maximal order in G/N5 . Therefore,
we set A(G,N5 ,1N5 ) = N4 . We now use fact 5 to compute Cl(G, N5 , 1N5 ). It turns
out that K = {hxi4 x5 , x6 i | 0 ≤ i ≤ 4} ∪ {hx4 , x6 i}. Moreover, hx5 , x6 i, hx4 , x6 i,
hx−1
4 x5 , x6 i are the only subgroups in K which are distinct up to conjugacy in G.
Consider the linear characters ϕ1 , ϕ2 , ϕ3 on N4 given as follows:
ϕ1 : x4 7→ ξ5 , x5 7→ 1, x6 7→ 1
ϕ2 : x4 7→ 1, x5 7→ ξ5 , x6 →
7 1
ϕ3 : x4 7→ ξ5 , x5 7→ ξ5 , x6 7→ 1,

24

where ξ5 is a primitive 5th root of unity. Clearly the kernels of ϕ1 , ϕ2 and ϕ3 are
hx5 , x6 i, hx4 , x6 i and hx−1
4 x5 , x6 i respectively. Also we have IG (ϕ1 ) = IG (ϕ2 ) =
IG (ϕ3 ) = N4 . Hence, the directed tree is as follows:
(N4 , N4 , ϕ1 ) (N4 , N4 , ϕ2 ) (N4 , N4 , ϕ3 )

(G, N5 , 1N 5 )

Figure 6: GN5
The three leaves (N4 , N4 , ϕ1 ), (N4 , N4 , ϕ2 ), (N4 , N4 , ϕ3 ), respectively, yield Shoda
pairs (hx4 , x5 , x6 i, hx5 , x6 i), (hx4 , x5 , x6 i, hx4 , x6 i) and (hx4 , x5 , x6 i, hx−1
4 x5 , x6 i) of
G.
Construction of GN6 : Recall that N6 = h1i. Note that N5 is an abelian normal
subgroup of maximal order in G. Therefore, we set A(G,N6 ,1N6 ) = N5 . As N5 /N6 is
cyclic, by fact 2,
Cl(G, N6 , 1N6 ) = {(G1 , N5 , ϕ1 )},
where ϕ1 : N5 → C maps x6 to ξ5 , and G1 = IG (ϕ1 ) = hx3 , x4 , x5 , x6 i. As
G1 6= N5 , we further need to compute d.c.c. of (G1 , N5 , ϕ1 ). It is observed that
hx5 , x6 i/ ker ϕ1 is an abelian normal subgroup of maximal order in G1 / ker ϕ1 . Set
A(G1 ,N5 ,ϕ1 ) = hx5 , x6 i. We now use fact 5 to compute Cl(G1 , N5 , ϕ1 ). It turns out
that there are 5 subgroups in K, namely hx5 xi6 i, 0 ≤ i ≤ 4, and all of them are
conjugate in G1 . Consider the linear character ϕ2 on hx5 , x6 i which maps x5 to 1
and x6 to ξ5 . Hence
Cl(G1 , N5 , ϕ1 ) = {(G2 , hx5 , x6 i, ϕ2 )},
where G2 = IG1 (ϕ2 ) = hx5 , x6 , x3 x24 i. Again G2 6= hx5 , x6 i, and we compute
Cl(G2 , hx5 , x6 i, ϕ2). Now G2 / ker ϕ2 is abelian. Therefore, using facts 5 and 6,
we obtain that
Cl(G2 , hx5 , x6 i, ϕ2) = {(G2 , G2 , ϕ3 ), (G2 , G2 , ϕ4 )},
where ϕ3 and ϕ4 are linear characters on G2 with kernel hx3 x24 x36 , x35 i and hx35 i
respectively. The process stops here and the corresponding tree is as follows:

25

(G2 , G2 , ϕ3 ) (G2 , G2 , ϕ4 )

(G2 , hx5 , x6 i, ϕ2 )

(G1 , N5 , ϕ1 )

(G, N6 , 1N6 )

Figure 7: GN6
The leaves (G2 , G2 , ϕ3 ) and (G2 , G2 , ϕ4 ) of GN6 correspond to the Shoda pairs
(hx5 , x6 , x3 x24 i, hx3 x24 x36 , x35 i) and (hx5 , x6 , x3 x24 i, hx35 i) of G respectively. It turns
out that the collection of Shoda pairs corresponding to all the GNi , 1 ≤ i ≤ 6, is a
complete irredundant set of Shoda pairs of G. Futhermore, if (H, K) is any of the
Shoda pairs constructed with this process, then, from Theorem 4, it follows that

1/2, if (H, K)=(hx5 , x6 , x3 x24 i, hx3 x24 x36 , x35 i) or (hx5 , x6 , x3 x24 i, hx35 i);
α(G,H,K) = 1,
otherwise.

7.2

Example 2

Consider the group G generated by a, b, c, d, e, f with the following defining
relations:
a2 =b3 =c3 =d3 =1, a−1 ba=b−1 , a−1 ca=c−1 , a−1 da=d, b−1 cb=cd, b−1 db=d, c−1 dc=d.
There are 8 normal subgroups of G given as follows: N1 =G, N2 =hb, ci, N3 =hc, di,
N4 =hcb, c−1 b−1 i, N5 =hcb−1 , c−1 bi, N6 =hb, di, N7 =hdi and N8 = h1i. As before, the
directed tree GN1 is just the vertex (G, G, 1G ), which gives Shoda pair (G, G). As
G/N2 is cyclic, by fact 3, the tree corresponding to N2 is as given below:
(G, G, ϕ)

(G, N2 , 1N2 )

Figure 8: GN2
where ϕ can be taken as any linear character on G with kernel N2 , which gives
Shoda pair (G, N2 ) of G.

26

For 3 ≤ i ≤ 6, N2 /Ni is an abelian normal subgroup of maximal order in G/Ni .
Hence we set A(G,Ni ,1Ni ) = N2 . It turns out that A(G,Ni ,1Ni ) /Ni is cyclic and maximal among all the abelian subgroups of G/Ni . Therefore by fact 2, GNi , 3 ≤ i ≤ 6,
can be described as follows:

(hb, ci, hb, ci, ϕ1 )

(G, N3 , 1N3 )

Figure 9: GN3

(hb, ci, hb, ci, ϕ2 )

(hb, ci, hb, ci, ϕ3 )

(G, N4 , 1N4 )

(G, N5 , 1N5 )

Figure 10: GN4

Figure 11: GN5

(hb, ci, hb, ci, ϕ4 )

(G, N6 , 1N6 )

Figure 12: GN6

where ϕ1 , ϕ2 , ϕ3 , ϕ4 are any linear characters on hb, ci with kernel N3 , N4 , N5 , N6
respectively. The above trees yield Shoda pairs (hb, ci, N3 ), (hb, ci, N4 ), (hb, ci, N5),
(hb, ci, N6 ) of G.
Next, it turns out that Cl(G, N7 , 1N7 ) is empty. Therefore, GN7 is just the vertex
(G, N7 , 1N7 ), which does not give any Shoda pair.
It now remains to construct the directed tree GN8 . Observe that N3 /N8 is an
abelian normal subgroup of maximal order in G/N8 . Hence we can set A(G,N8 ,1N8 ) =
N3 . In view of fact 5, we have Cl(G, N8 , 1N8 ) = {(G1 , N3 , ϕ)}, where ϕ is a
linear character on N3 with kernel hci and G1 =IG (ϕ)=ha, c, di. As G1 6= N3 , we
further compute Cl(G1 , N3 , ϕ). Now observe that G1 / ker ϕ = G1 /hci is abelian
and therefore facts 5 & 6 yield
Cl(G1 , N3 , ψ) = {(G1 , G1 , ϕ1 ), (G1 , G1 , ϕ2 )},
where ϕ1 and ϕ2 can be taken to be any linear characters of G1 with kernel hci and
ha, ci respectively. The process stops here and the corresponding tree is as follows:
(G1 , G1 , ϕ1 ) (G1 , G1 , ϕ2 )

(G1 , hc, di, ϕ)

(G, N8 , 1N8 )

Figure 13: GN8
and it yields Shoda pairs (ha, c, di, hci) and (ha, c, di, ha, ci) of G. Again, it turns
out that the collection of Shoda pairs constructed with this process is a complete
and irredundant set of Shoda pairs of G.
27

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29

