arXiv:1705.06456v1 [] 18 May 2017

Groups whose Chermak-Delgado lattice is a quasi-antichain∗
Lijian An
Department of Mathematics, Shanxi Normal University
Linfen, Shanxi 041004, P. R. China
April 1, 2018

Abstract
A quasiantichain is a lattice consisting of a maximum, a minimum, and the
atoms of the lattice. The width of a quasiantichian is the number of atoms. For
a positive integer w (≥ 3), a quasiantichain of width w is denoted by Mw . In [3],
it is proved that Mw can be as a Chermak-Delgado lattice of a finite group if and
only if w = 1 + pa for some positive integer a. Let t be the number of abelian
atoms in CD(G). If t > 2, then, according to [3], there exists a positive integer
b such that t = pb + 1. The converse is still an open question. In this paper, we
proved that a = b or a = 2b.
Keywords finite p-groups
generated torsion module

Chermak-Delgado lattice

quasi-antichain finitely

2000 Mathematics subject classification: 20D15.

Chermak and Delgado [5] defined a family of functions from the set of subgroups of a
finite group into the set of positive integers. They then used these functions to obtain
a variety of results, including a proof that every finite group G has a characteristic
abelian subgroup N such that |G : N | ≤ |G : A|2 for any abelian A ≤ G. ChermakDelgado measures are values of one of these functions. For any subgroup H of G,
the Chermak-Delgado measure of H (in G) is denoted by mG (H), and defined as
mG (H) = |H||CG (H)|. The maximal Chermak-Delgado measure of G is denoted by
m∗ (G). That is , m∗ (G) = max{mG (H) | H ≤ G}. In [6], Isaacs showed the following
theorem.
Theorem 1. ([6]) Given a finite group G, let CD(G) = {H | mG (H) = m∗ (G)}. Then
(1) CD(G) is a lattice of subgroups of G.
(2) If H, K ∈ CD(G), then hH, Ki = HK.
(3) If H ∈ CD(G), then CG (H) ∈ CD(G) and CG (CG (H)) = H.
By Theorem 1, Chermak-Delgado lattice of a finite groups is always a self-dual
lattice. It is natural to ask a question: which types of self-dual lattices can be used
as Chermak-Delgado lattices of finite groups. Some special cases of the above question
∗

This work was supported by NSFC (No. 11471198)

1

are proposed and solved. In [2], it is proved that, for any integer n, a chain of length
n can be a Chermak-Delgado lattice of a finite p-group. In [1], general conclusions are
given.
Theorem 2. ([1]) If L is a Chermak-Delgado lattice of a finite p-group G such that both
G/Z(G) and G′ are elementary abelian, then are L+ and L++ , where L+ is a mixed
3-string with center component isomorphic to L and the remaining components being
m-diamonds (a lattice with subgroups in the configuration of an m-dimensional cube),
L++ is a mixed 3-string with center component isomorphic to L and the remaining
components being lattice isomorphic to Mp+1 (a quasiantichain of width p + 1, see the
following definition).
A quasiantichain is a lattice consisting of a maximum, a minimum, and the atoms
of the lattice. The width of a quasiantichian is the number of atoms. For a positive
integer w (≥ 3), a quasiantichain of width w is denoted by Mw . In [3], it is proved that
Mw can be as a Chermak-Delgado lattice of a finite group if and only if w = 1 + pa
for some positive integer a. According to [3], if Mw is a Chermak-Delgado lattice of
a finite group G, then G is nilpotent of class 2. Moreover, we may choose G to be
a p-group of class 2 without loss of generality. Let M1 , M2 , . . . , Mw be all atoms of
CD(G). Then G has the following properties:
(P1) Both G/Z(G) and G′ are elementary abelian;
(P2) Mi /Z(G) ∼
= Mj /Z(G) and G/Z(G) = Mi /Z(G) × Mj /Z(G) for i 6= j;
(P3) Let M1 = hx1 , x2 , . . . , xn iZ(G) and M2 = hy1 , y2 , . . . , yn iZ(G) such that
M1 /Z(G) and M2 /Z(G) are elementary abelian groups of order pn . Then, for k ≥ 3,
Mk = hx1 y1′ , x2 y2′ , . . . , xn yn′ iZ(G) where M2 = hy1′ , y2′ , . . . , yn′ iZ(G). Moreover, there is
(k)
an invertible matrix Ck = (cij )n×n over Fp such that
yj′ ≡

n
Y

c

(k)

yi ij (mod Z(G)), j = 1, 2, . . . , n.

i=1

In this paper, Ck is called the characteristic matrix of Mk relative to (x1 , x2 , . . . , xn )
and (y1 , y2 , . . . , yn ).
Let G be a p-group with CD(G) a quasi-antichain of width w ≥ 3. Let t be the
number of abelian atoms in CD(G). If t > 2, then, according to [3], there exists a
positive integer b such that t = pb + 1. The converse is still an open question. At the
end of [3], such questions are proposed definitely:
(Q1) Which values of t are possible in quasi-antichain Chermak-Delgado lattices of
width w = pa + 1 when a > 1?
(Q2) Are there examples of groups G with G ∈ CD(G) and CD(G) a quasi-antichain
where t = 0 and p ≡ 1 modulo 4?
This paper answer above questions completely. It is amazing that there are only
two possible relations: a = b or a = 2b. We also prove that a | n, which is another appli2

cation about the decomposition of a finitely generated torsion module over a principal
ideal domain. Examples related to (Q1) and (Q2) are given.
Throughout the article, let p be a prime and G be a finite p-group with properties
(P1), (P2) and (P3).
Theorem 3. Suppose that there are totally t > 2 abelian atoms in CD(G), which is a
quasi-antichain of width w. Then there are positive integers a and b such that w = pa +1
and t = pb + 1, where a = b or a = 2b. If |G/Z(G)| = p2n , then a | n.
Proof Without loss of generality, we may assume that M1 , M2 and M3 are abelian
atoms. For convenience, the operation of the group G is replaced to be addition.
According to (P3), M3 = hx1 + y1′ , x2 + y2′ , . . . , xn + yn′ iZ(G) such that
yj′ ≡

n
Y

(3)

cij yi (mod Z(G)), j = 1, 2, . . . , n.

i=1

(3)

where C3 = (cij )n×n is the characteristic matrix of Mk relative to (x1 , x2 , . . . , xn ) and
(y1 , y2 , . . . , yn ). Replacing y1 , y2 , . . . , yn with y1′ , y2′ , . . . , yn′ respectively, we may assume
that C3 = In .
Let zij = [xi , yj ] and Z = (zij )n×n . Then G′ = hzij | i, j = 1, 2, . . . , ni. We have the
following formalized calculation:
[((x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )Ck )T , (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )Ck′ ]
= [(x1 , x2 , . . . , xn )T + CkT (y1 , y2 , . . . , yn )T , (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )Ck′ ]
=

[(x1 , x2 , . . . , xn )T , (y1 , y2 , . . . , yn )Ck′ ] + [CkT (y1 , y2 , . . . , yn )T , (x1 , x2 , . . . , xn )]

=

ZCk′ + CkT (−Z T ) = ZCk′ − (ZCk )T .

Hence CG (Mk ) = Mk′ if and only if (ZCk )T = ZCk′ . In particular, Mk is abelian if
and only if ZCk is symmetric. Since M3 is abelian, Z T = Z.
Let V = {C ∈ Mn (Fp ) | ∃ C ′ ∈ Mn (Fp ) s.t. C T Z = ZC ′ }. It is straightforward to
prove that V is a linear space over Fp . Hence |V| = pa for some positive integer a. In
the following, we prove that V is also a field.
Assume that On 6= C ∈ V and C T Z = ZC ′ . Let
(s1 , s2 , . . . , sn ) = (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )C
and
(t1 , t2 , . . . , tn ) = (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )C ′ .
Then, by the above formalized calculation, [(s1 , s2 , . . . , sn )T , (t1 , t2 , . . . , tn )] = On . Let
N1 = hs1 , s2 , . . . , sn iZ(G) and N2 = ht1 , t2 , . . . , tn iZ(G). Then N2 ≤ CG (N1 ). Hence
|N1 ||CG (N1 )| ≥ |N1 ||N2 | = |M1 |2 = m∗ (G). It implies that N1 ∈ CD(G), and, there is
some k ≥ 3 such that N1 = Mk . Hence V \ {On } is the set of Ck where 3 ≤ k ≤ w.
3

Since Ck is invertible, V is a division algebra. By Wedderburn’s little theorem, V is
also a finite field. Now we know that w − 2 = pa − 1. hence w = pa + 1. Let A be a
a
primitive element of V. Then V = {On , A, A2 , . . . , Ap −1 = In }.
Next we study A applying decomposition of a finitely generated torsion module over
a principal ideal domain. M2 /Z(G) can be regarded as a n-dimensional vector space
V2 over Fp , in which (y1 Z(G), y2 Z(G), . . . , yn Z(G)) is a base for V2 . At this time,
A = (aij ) is a matrix of a single transformation A relative to the given base, where
n
Y
A(yj Z(G)) = ( aij yi )Z(G), j = 1, 2, . . . , n.
i=1

We can make V2 an Fp [λ]-module by defining the action of any polynomial g(λ) =
b0 + b1 λ + · · · + bm λm on any vector y ∈ V2 as
g(λ)y = b0 + b1 (Ay) + · · · + bm (Am y).
Let mA (λ) be the minimal polynomial of A. Then V2 is a torsion Fp [λ]-module of
exponent mA (λ). Obviously, mA (x) is irreducible with degree a. By fundamental
structure theorem for finitely generated torsion modules over a principle ideal domain,
V2 is a direct sum of cyclic module:
V2 = Fp [λ]v1 ⊕ Fp [λ]v2 ⊕ · · · ⊕ Fp [λ]vr
It is easy to see that annvi = mA (λ) where i = 1, 2, . . . , r. Let mA (λ) = k0 + k1 λ +
· · · + ka−1 λa−1 − ka λa . Then n = ra and
v1 , Av1 , . . . , Aa−1 v1 , v2 , Av2 , . . . , Aa−1 v2 , . . . , vr , Avr , . . . , Aa−1 vr
is a set of generators for M2 . Assume that (v1 , v2 , . . . , vn ) = (y1 , y2 , . . . , yn )S. Let
(u1 , u2 , . . . un ) = (x1 , x2 , . . . , xn )S where S = (sij ) is an invertible matrix. Then the
characteristic matrix of Mk relative to (u1 , u2 , . . . , un ) and (v1 , v2 , . . . , vn ) is S −1 Ck S.
Matrices related in this way are said to be similar.
Without loss of generality, we may assume that A is the characteristic matrix relative to (v1 , Av1 , . . . , Aa−1 vr ). It is clear that A has the form


A1 Oa . . . Oa
 Oa A2 . . . Oa 


A= .
..
..  ,
..
 ..
.
.
. 
Oa

0

...

Ar

where Ai is the companion matrix of mA (λ). That is,

0 0 ...
k0
 1 0 ...
k1

Ai = B =  . .
..
.
.
.
.
 .
.
.
.
0 ...
1 ka−1
4





,


and A = Diag(B, B, . . . , B).
If AT Z = ZA′ , then A′ T Z = ZA since Z T = Z.
there are 1 ≤ k ≤ pa − 1 such that AT Z = ZAk . Let

Z11 Z12 . . . Z1r
 Z21 Z22 . . . Z2r

Z= .
..
..
..
 ..
.
.
.
Zr1 Zr2 . . .

Zrr

It follows that A′ ∈ V. Hence





,


T for 1 ≤ i ≤ j ≤ r. It follows
where Zij are a × a matrices. Since Z T = Z, Zji = Zij
from AT Z = ZAk that



(1)

Zij

 (2)
 Zij
Let Zij = 
 ..
 .
(a)
Zij

B T Zij = Zij B k

(1)

B T Zji = Zji B k

(2)





. Then


(2)





B Zij = 





Zij

(3)

Zij
..
.

T

(1)

(2)

(a)

k0 Zij + k1 Zij + · · · + ka−1 Zij







(3)

By (1) and (3),
(2)

(1)

(3)

(2)

(a)

(a−1)

Zij = Zij B k , Zij = Zij B k , . . . , Zij = Zij
(1)

(2)

(a)

Bk

(a)

k0 Zij + k1 Zij + · · · + ka−1 Zij = Zij B k

(4)
(5)

Hence
(1)

Zij (k0 + k1 B k + · · · + ka−1 B (a−1)k − B ak ) = 0

(6)

(1)

That is, Zij mA (B k ) = 0. We claim that mA (B k ) = 0. Otherwise, mA (Ak ) 6= 0. Since
mA (Ak ) ∈ V and V is a field, mA (Ak ) is invertible. Hence mA (B k ) is also invertible.
(1)
Notice that if Zij = 0 then Zij = Oa by (4) and (5). Hence we may choose Zij such
(1)

(1)

that Zij 6= 0. In this case, Zij mA (B k ) 6= 0, a contradiction. Thus mA (B k ) = 0 and
a
2
mA (Ak ) = 0. Since Ap , Ap , . . . , Ap = A are all zero points of mA (λ), there exists a
1 ≤ e ≤ a such that k = pe .
e
It is easy to see that (Am )T Z = ZAmp if and only if (pa − 1) | m(pe − 1). Let
W = {C ∈ V | C T Z = ZC}. Then W \ {On } = {Am ∈ V | (pa − 1) | m(pe − 1)}. Hence

5

pa −1

W \ {On } is a cyclic group generated by A (pa −1,pe −1) of order (pa − 1, pe − 1) = p(a,e) − 1.
Let b = (a, e). Notice that |W| = t − 1. It is easy to see that t = pb + 1. By (1),
e

(B p )T Zij = Zij B p

2e

(7)

By (2),
e

(B p )T Zij = Zij B

(8)

Hence (pa − 1) | (p2e − 1). Thus a | 2e. It follows that a | 2b. Hence a = e = b or
a = 2e = 2b.

Remark 4. Let F be a field containing pa elements and F ∗ be the multiply group of
F . Then F ∗ is cyclic with order pa − 1. Let F ∗ = hbi and B : f 7→ bf be linear
transformation over F , where F is regarded as a linear space over field Fp . Of course,
the order of B is pa −1. Let p(x) be the minimal polynomial of B. It follows from CayleyHamilton theorem that deg(p(x)) = r ≤ a. Let W = {f (B) f (x) ∈ Fp [x]} = {f1 (B) |
f1 (x) ∈ Fp [x], deg(f1 ) < r}. Then dimW = deg(p(x)) = r and hence |W | = pr . On the
a
other hand, {1, B, B 2 , . . . , B p −1 } ⊆ W and hence |W | ≥ pa . So we get r = a. Let the
minimal polynomial of B is
p(x) = xa − ka−1 xa−1 − · · · − k1 x − k0 .
Then a matrix of B is Frobenius form

0 0
 1 0

B= . .
..
 ..
0 ...

...
...
..
.

k0
k1
..
.

1

ka−1





.


Lemma 5. Let B be the matrix introduced in Remark 4. Z = (zij ) is a a × a matrix.
If B T Z = ZB, Then Z is symmetric.
Proof By calculation, the (u, v)th entry of B T Z is zu+1,v , while the (u, v)th entry of
ZB is zu,v+1 , where 1 ≤ u, v ≤ a − 1. Hence zu+1,v = zu,v+1 where 1 ≤ u, v ≤ a − 1.
Then, for u ≤ v, we have zu,v+1 = zu+1,v = zu+2,v−1 = · · · = zv,u+1 = zv+1,u . Hence Z
is symmetric.
Theorem 6. Suppose that ???
Proof Let G be generated by {x1 , x2 , . . . , xn , y1 , y2 , . . . , yn } with defining relationships xpi = yip = 1 and [xi , xj ] = [yi , yj ] = 1 for all i, j such that 1 ≤ i, j ≤
n, [x(u−1)a+i , y(v−1)a+j ] = z(u−1)a+i,(v−1)a+j for 1 ≤ u, v ≤ r and 1 ≤ i, j ≤ a,
z(u−1)a+1,(v−1)a+j ∈ Z(G) for 1 ≤ u ≤ v ≤ r and 1 ≤ j ≤ a. For convenience,

6

we use addition operation to replace mutiplication operation of G. We also use the
following notations (where 1 ≤ u, v ≤ r and 1 ≤ i, j, k ≤ a):
(k)
Zuv
:= (z(u−1)a+k,(v−1)a+1 , z(u−1)a+k,(v−1)a+2 , . . . , z(u−1)a+k,(v−1)a+a ),



Zuv



= (z(u−1)a+i,(v−1)a+j ) = 



(1)

Zuv
(2)
Zuv
..
.
(3)

Zuv








 and Z = 





Z11 Z12 . . .
Z21 Z22 . . .
..
..
..
.
.
.
Zr1 Zr2 . . .

Z1r
Z2r
..
.
Zrr

(k)





.


(1)

Using above notations, we continue to give defining relationships Zuv = Zuv B k−1 for
T for 1 ≤ v < u ≤ r. It is easy to see that
1 ≤ u ≤ v ≤ r and 2 ≤ k ≤ a, Zuv = Zvu
G′ = Z(G) = hz(u−1)a+1,(v−1)a+j | 1 ≤ u ≤ v ≤ r, 1 ≤ j ≤ ai is elementary abelian of
r+1
r+5
order p 2 n . Hence |G| = p 2 n .
(k)
(1)
Since Zuv = Zuv B k−1 for 1 ≤ u ≤ v ≤ r and 2 ≤ k ≤ a, B T Zuv = Zuv B
for 1 ≤ u ≤ v ≤ r. By Lemma 5, Zuv is symmetric for 1 ≤ u ≤ v ≤ r. Hence
T
T
T = Z
Zuv = Zvu
vu for 1 ≤ v < u ≤ r. Moreover Zuv = Zvu = Zuv = Zvu for all
1 ≤ u, v ≤ r and Z T = Z. Let X = hx1 , x2 , . . . , xn iZ(G) and Y = hy1 , y2 , . . . , yn iZ(G).
Assertion 1: CG (x) = X for all x ∈ X \ Z(G).
Q
Let x = ni=1 ci xi +z where z ∈ Z(G). Write Ck = (c(k−1)a+1 , c(k−1)a+2 , . . . , cka ) for
1 ≤ k ≤ a. Since x 6∈ Z(G), there exists a k0 such that Ck0 6= (0, 0, . . . , 0). Exchanging
x(k0 −1)a+1 , y(k0 −1)a+1 , . . . , xk0 a , yk0 a and x(r−1)a+1 , y(r−1)a+1 , . . . , xra , yra , we have Cr 6=
(0, 0, . . . , 0). Let Hi = h[x, y(i−1)a+j ] | 1 ≤ j ≤ ai for 1 ≤ i ≤ r. We will prove that
H1 +H2 +· · ·+Hv is of order pva . Use induction, we may assume that H1 +H2 +· · ·+Hv−1
is of order p(v−1)a . By calculation,
([x, y(v−1)a+1 ], . . . , [x, yva ])
n
X
=
ck ([xk , y(v−1)a+1 ], . . . , [xk , yva ])
k=1

=
=
=
=
=

r X
a
X

u=1 i=1
r X
a
X

u=1 i=1
r X
a
X

c(u−1)a+i ([x(u−1)a+i , y(v−1)a+1 ], . . . , [x(u−1)a+i , yva ])
(i)
c(u−1)a+i Zuv

(1) i−1
c(u−1)a+i Zuv
B

u=1 i=1
a
r
X
X
(1)
c(u−1)a+i B i−1 )
Zuv
(
u=1
i=1
a
r−1
X
X
(1)
c(u−1)a+i B i−1 ) +
Zuv (
u=1
i=1

7

a
X

(1)
Zrv
(

i=1

c(r−1)a+i B i−1 )

P
P
Since Cr 6= (0, 0, . . . , 0), ( ai=1 c(r−1)a+i B i−1 ) 6= Oa . By Remark 4, ( ai=1 c(r−1)a+i B i−1 )
(1) P
is invertible. Hence Zrv ( ai=1 c(r−1)a+i B i−1 ) is of rank a. It follows that (H1 + H2 +
· · · + Hv−1 ) ∩ Hv = 0. Thus H1 + H2 + · · · + Hv is of order pva .
By above discussion, H1 +H2 +· · ·+Hv is of order pn . Hence |[x, G]| = pn . It follows
r+3
r+3
that |CG (x)| = |G|/pn = p 2 n . Since |X| = p 2 n and X ≤ CG (x), CG (x) = X.
Similarly, we have CG (y) = Y for all y ∈ Y \ Z(G). Then CG (X) = X and
CG (Y ) = Y , yielding mG (G) = mG (X) = mG (Y ) = p(r+3)n .
Assertion 2: m∗ (G) = p(r+3)n .
Otherwise, by the dual-property of CD-lattice, there exists H ∈ CD(G) such that
(r+3)
(r+1)
n
2
H < G and |H| > p 2 n . Since |H ∩ X| = |H||X|
= |Z(G)|, there exists
|HX| > p
x ∈ H ∩ X \ Z(G). Hence CG (H) ≤ CG (x) = X. Similarly, we have CG (H) ≤ Y .
Hence CG (H) = Z(G) and mG (H) < mG (G), a contradiction.
Above discussion also gives that CD(G) is a quasi-antichain, in which every atom
(r+3)
is of order p 2 n .
Assertion 3: w = t = 1 + pa .
Now we have G, Z(G), X, Y ∈ CD(G). Let M be an atom different from X and Y .
Then, by the same reason given in Theorem 3, we may let M = hw1 , w2 , . . . , wn i and
N = CG (M ) = hv1 , v2 , . . . , vn i where
(w1 , w2 , . . . , wn ) = (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )C,
(v1 , v2 , . . . , vn ) = (x1 , x2 , . . . , x3n ) + (y1 , y2 , . . . , yn )D.
Since [M, N ] = 0, C T Z = ZD. Let

C11 C12 . . . C1r
 C21 C22 . . . C2r

C= .
..
..
..
 ..
.
.
.
Cr1 Cr2 . . . Crr
Then

r
X
k=1

T
Cki
Zkj

=









 and D = 


r
X

D11 D12 . . .
D21 D22 . . .
..
..
..
.
.
.
Dr1 Dr2 . . .

Zik Dkj for 1 ≤ i, j ≤ r.

D1r
D2r
..
.
Drr





.


(9)

k=1

Taking i 6= j in Equation (9) and compiling two sides, we get Cik = Oa for i 6= k,
Dkj = Oa for k 6= j, and
CiiT Zij = Zij Djj for 1 ≤ i, j ≤ r and i 6= j.

(10)

Taking i = j in Equation (9) and compiling two sides, we get
CiiT Zii = Zii Dii for 1 ≤ i ≤ r.

(11)

Notice that Zij and Z11 have no essential difference. Thus Cii = Cjj and Dii = Djj for
i 6= j.
8

Let V = {U ∈ Ma (Fp ) | ∃ V ∈ Ma (Fp ) s.t. U T Z11 = Z11 V }. Similar to the proof
in Theorem 3, V is a division algebra. Hence V has at most pa elements (otherwise,
there exists a non-zero matrix in which the elements of the first row are all zero, a
contradiction). Notice that B T Z11 = Z11 B. Hence B k ∈ V where 1 ≤ k ≤ pa − 1. it
a
follows that V = {On , B, B 2 , . . . , B p −1 = In }.
Therefore C = D = Diag(B k , B k , . . . , B k ) for some 1 ≤ k ≤ pa − 1. Hence w = t =
pa + 1.

Theorem 7. Suppose that a, b and r are positive integers such that a = 2b and r ≥ 3.
Let n = ar. Then there exists a group G such that |G/Z(G)| = p2n , and CD(G) is a
quasi-antichain of width w = pa + 1, in which the number of abelian atoms is t = pb + 1.
Proof Let G be generated by {x1 , x2 , . . . , xn , y1 , y2 , . . . , yn } with defining relationships
similar to that in Theorem 6. The followings are different defining relationships from
(k)
that in Theorem 6: [x(u−1)a+i , y(u−1)a+j ] = 1 for 1 ≤ u ≤ r and 1 ≤ i, j ≤ a, Zuv =
b
(1)
Zuv B (k−1)p for 1 ≤ u < v ≤ r and 2 ≤ k ≤ a.
In this case, G′ = Z(G) = hz(u−1)a+1,(v−1)a+j | 1 ≤ u < v ≤ r, 1 ≤ j ≤ ai is
r+3
r−1
elementary abelian of order p 2 n . Hence |G| = p 2 n .
b
b
(k)
(1)
Since Zuv = Zuv B (k−1)p for 1 ≤ u < v ≤ r and 2 ≤ k ≤ a, B T Zuv = Zuv B p for
1 ≤ u < v ≤ r. In this case, Zuv is not symmetric and Zuv 6= Zvu for all 1 ≤ v < u ≤ r.
Z T = Z is still hold. Let X = hx1 , x2 , . . . , xn iZ(G) and Y = hy1 , y2 , . . . , yn iZ(G).
Similar to Assertion 1 in the proof of Theorem 6, we have |[x, Y ]| ≥ p(r−1)a for all
x ∈ X \ Z(G). It follows that |CY (x)| ≤ |Y |/p(r−1)a = pa |Z(G)|.
Similarly, |CX (y)| ≤ pa |Z(G)| for all y ∈ Y \ Z(G). It is easy to check that
CG (X) = X and CG (Y ) = Y , yielding mG (G) = mG (X) = mG (Y ) = p(r+1)n .
Assertion (1): m∗ (G) = p(r+1)n .
Otherwise, by the dual-property of CD-lattice, there exists H ∈ CD(G) such that
(r−1)
(r+1)
n
2
= |Z(G)|, there exists
H < G and |H| > p 2 n . Since |H ∩ X| = |H||X|
|HX| > p
x ∈ H ∩ X \ Z(G). Hence CG (H) ≤ CG (x) = XCY (x). Similarly, there exists y ∈ H ∩
Y \ Z(G). Hence CG (H) ≤ CG (y) = Y CX (y). It follows that CG (H) ≤ CX (y)CY (x).
Hence |CG (H)| ≤ p2a |Z(G)|. Obviously CG (H) > Z(G). Hence there exists 1 6=
x′ y ′ ∈ CG (H) \ Z(G), where x′ ∈ CX (y), y ′ ∈ CY (x). Whatever, we have |H| ≤
r+1
|CG (x′ y ′ )| ≤ |G : [G, x′ y ′ ]| ≤ p 2 n+a . Hence mG (H) = |H||CG (H)| ≤ prn+3a ≤
p(r+1)n , a contradiction.
Assertion (2): CD(G) is a quasi-antichain.
Otherwise, by the dual-property of CD-lattice, there exists H ∈ CD(G) such that
(r+1)
H < G and |H| > p 2 n . By Assertion (1), n = 3a and there exists x ∈ H ∩ X \ Z(G)
and y ∈ H ∩ Y \ Z(G) such that CG (H) = CX (y)CY (x), where |CX (y)| = |CY (x)| =
pa |Z(G)|. Take x′ ∈ CX (y) \ Z(G) and y ′ ∈ CY (x) \ Z(G). Then H ≤ CX (y ′ )CX (x′ ) ≤
(r+1)
p2a |Z(G)| < p 2 n , a contradiction.
9

Assertion (3): w = pa + 1 and t = pb + 1.
Let M be an atom different from X and Y . Then, by the same reason given in
Theorem 3, we may let M = hw1 , w2 , . . . , wn i and N = CG (M ) = hv1 , v2 , . . . , v3n i
where
(w1 , w2 , . . . , wn ) = (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )C,
(v1 , v2 , . . . , v3 ) = (x1 , x2 , . . . , x3n ) + (y1 , y2 , . . . , y3 )D.
We also have C T Z = ZD.

C11 C12
 C21 C22

C= .
..
 ..
.
Then

Let
...
...
..
.

C1r
C2r
..
.

Cr1 Cr2 . . .

Crr

r
X
k=1

T
Zkj =
Cki









 and D = 


r
X

D11 D12 . . .
D21 D22 . . .
..
..
..
.
.
.
Dr1 Dr2 . . .

Zik Dkj for 1 ≤ i, j ≤ r.

D1r
D2r
..
.
Drr





.


(12)

k=1

Taking i 6= j in Equation (12) and compiling two sides, we get Cik = Oa for i 6= k,
Dkj = Oa for k 6= j, and
CiiT Zij = Zij Djj for 1 ≤ i, j ≤ r and i 6= j.

(13)

Notice that Zij , Zik and Zki have no essential difference for k 6= i, j. Thus Cii = Ckk and
Djj = Dkk for k 6= i, j. Since r ≥ 3, C11 = C22 = · · · = Crr and D11 = D22 = · · · = Drr .
Let V = {U ∈ Ma (Fp ) | ∃ V ∈ Ma (Fp ) s.t. U T Z12 = Z12 V }. Similar to the proof
in Theorem 3, V is a division algebra. Hence V has at most pa elements (otherwise,
there exists a non-zero matrix in which the elements of the first row are all zero, a
b
contradiction). Notice that B T Z12 = Z12 B p . Hence B k ∈ V where 1 ≤ k ≤ pa − 1. It
a
follows that V = {On , B, B 2 , . . . , B p −1 = In }.
b
b
b
Therefore C = Diag(B k , B k , . . . , B k ) and D = Diag(B kp , B kp , . . . , B kp ) for some
1 ≤ k ≤ pa − 1. Hence w = pa + 1. It is easy to see that M = N if and only if C = D
if and only if (pb + 1) | k. Hence the number of abelian atoms t = pb + 1.

Theorem 8. Suppose that a and r are positive integers such that r ≥ 3. Let n = ar.
Then there exists a group G such that |G/Z(G)| = p2n , and CD(G) is a quasi-antichain
of width w = pa + 1, in which the number of abelian atoms is 1 or 2 for p = 2 or p 6= 2
respectively.
Proof Let G be generated by {x1 , x2 , . . . , xn , y1 , y2 , . . . , yn } with defining relationships
xpi = yip = 1 and [xi , yj ] = 1 for all i, j such that 1 ≤ i, j ≤ n, [x(u−1)a+i , x(u−1)a+j ] = 1
for 1 ≤ u ≤ r and 1 ≤ i < j ≤ a, [x(u−1)a+i , x(v−1)a+j ] = z(u−1)a+i,(v−1)a+j for
1 ≤ u < v ≤ r and 1 ≤ i, j ≤ a, [yi , yj ] = [xj , xi ] for every i, j with 1 ≤ i < j ≤ n,
z(u−1)a+1,(v−1)a+j ∈ Z(G) for 1 ≤ u < v ≤ r and 1 ≤ j ≤ a. For convenience, we use
10

addition operation to replace mutiplication operation of G. We also use the following
notations (where 1 ≤ u, v ≤ r and 1 ≤ i, j, k ≤ a):
(k)
Zuv
:= (z(u−1)a+k,(v−1)a+1 , z(u−1)a+k,(v−1)a+2 , . . . , z(u−1)a+k,(v−1)a+a ),
 (1) 


Zuv
Z11 Z12 . . . Z1r
 (2) 
 Z21 Z22 . . . Z2r 
 Zuv 



Zuv = (z(u−1)a+i,(v−1)a+j ) = 
and
Z
=
 ..
..
..  .
..
 .. 

.
.
.
. 
 . 
(3)

Zr1 Zr2 . . .

Zuv

(k)

Zrr

(1)

Using above notations, we continue to give defining relationships Zuv = Zuv B k−1 for
T for 1 ≤ v < u ≤ r,
1 ≤ u < v ≤ r and 2 ≤ k ≤ a. It is easy to see that Zuv = −Zvu
G′ = Z(G) = hz(u−1)a+1,(v−1)a+j | 1 ≤ u < v ≤ r, 1 ≤ j ≤ ai is elementary abelian of
r+3
r−1
order p 2 n . Hence |G| = p 2 n .
Since [x(u−1)a+i , x(u−1)a+j ] = 1 for 1 ≤ u ≤ r and 1 ≤ i < j ≤ a, Zuu = Oa for
(k)
(1)
1 ≤ u ≤ r. Since Zuv = Zuv B k−1 for 1 ≤ u < v ≤ r and 2 ≤ k ≤ a, B T Zuv = Zuv B
for 1 ≤ u < v ≤ r. By Lemma 5, Zuv is symmetric for 1 ≤ u < v ≤ r. Hence
T
T = −Z
Zuv = −Zvu
vu for 1 ≤ v < u ≤ r. Moreover B Zuv = Zuv B for all 1 ≤ u, v ≤ r
and Z T = −Z. Let X = hx1 , x2 , . . . , xn iZ(G) and Y = hy1 , y2 , . . . , yn iZ(G).
Similar to Assertion 1 in the proof of Theorem 6, we have |[x, X]| ≥ p(r−1)a for all
x ∈ X \ Z(G). It follows that |CX (x)| ≤ |X|/p(r−1)a = pa |Z(G)|.
Similarly, |CY (y)| ≤ pa |Z(G)| for all y ∈ Y \Z(G). It is easy to check that CG (X) =
Y and CG (Y ) = X, yielding mG (G) = mG (X) = mG (Y ) = p(r+1)n .
Similar to Assertion (1) and Assertion (2) in the proof of Theorem 7, we have
= p(r+1)n and CD(G) is a quasi-antichain.
Let M be an atom different from X and Y . Then, by the same reason given in
Theorem 3, we may let M = hw1 , w2 , . . . , wn i and N = CG (M ) = hv1 , v2 , . . . , v3n i
where
(w1 , w2 , . . . , wn ) = (x1 , x2 , . . . , xn ) + (y1 , y2 , . . . , yn )C,
m∗ (G)

(v1 , v2 , . . . , vn ) = (x1 , x2 , . . . , xn )D + (y1 , y2 , . . . , yn ).
Since [M, N ] = 0, C T Z = ZD. Let

C11 C12 . . . C1r
 C21 C22 . . . C2r

C= .
..
..
..
 ..
.
.
.
Cr1 Cr2 . . . Crr









 and D = 



D11 D12 . . .
D21 D22 . . .
..
..
..
.
.
.
Dr1 Dr2 . . .

D1r
D2r
..
.
Drr





.


Similar to Assertion 3 in the proof of Theorem 6, we have C = D = Diag(B k , B k , . . . , B k )
for some 1 ≤ k ≤ pa − 1. Hence w = pa + 1. It is easy to see that M = N if and only if
CD = In if and only if (pa − 1) | 2k. Hence the number of abelian atoms is 1 or 2 for
p = 2 and p 6= 2 respectively.


11

Theorem 9. Suppose that p is and odd prime, a is odd, and r is positive integers such
that r ≥ 3. Let n = ar. Then there exists a group G such that |G/Z(G)| = p2n , and
CD(G) is a quasi-antichain of width w = pa + 1, in which the number of abelian atoms
is t = 0.
Proof Let G be generated by {x1 , x2 , . . . , xn , y1 , y2 , . . . , yn } with defining relationships
similar to that in Theorem 8. The unique different defining relationship is [yi , yj ] =
[xj , xi ]ν for every i, j with 1 ≤ i < j ≤ n, where ν is a fixed quadratic non-residue
module p.
Similar to the proof of Theorem 8, we have m∗ (G) = p(r+1)n , CD(G) is a quasiantichain, w = pa + 1, νC = D = Diag(B k , B k , . . . , B k ) for some 1 ≤ k ≤ pa − 1, and
M = N if and only if νB 2k = Ia .
By the definition of B, hB

pa −1
p−1

i = hlIa | 1 ≤ l ≤ p − 1i. Since ν is not a square,
a

−1
m pp−1

there exists an odd m such that B
= νIa .
a −1
pa −1
2k
a
If νB = Ia , then (p −1) | 2k +m p−1 . Notice that m and pp−1
= 1+p+· · ·+pa−1
are all odd. There is no integer k such that νB 2k = Ia . Hence the number of abelian
atoms is t = 0.


References
[1] AN, L., Brenna, J., Qu, H., Wilcox, E., Chermak-Delgado lattice extension theorems, Algebra comm., 2015, 43(5): 2201–2213.
[2] Brewster, B., Hauck, P., Wilcox, E., Groups whose Chermak-Delgado lattice is a
chain, Journal of Group Theory, 2014, 17(2): 253–265.
[3] Brewster, B., Hauck, P., Wilcox, E., Quasi-antichain Chermak-Delgado lattice of
finite groups, Arch. Math., 2014, 103: 301–311.
[4] Brewster, B., Wilcox, E., Some groups with computable Chermak-Delgado lattices,
Bulletin of the Australian Mathematical Society, 2012, 86(1): 29–40.
[5] Chermak, A., Delgado, A., A measuring argument for finite groups, Proceedings
of the American Mathematical Society, 1989, 107(4): 907–914.
[6] Isaacs, I. M., Finite Group Theory, American Mathematical Society, 2008.

12

