arXiv:1601.07667v1 [] 28 Jan 2016

Classification of group isotopes according to
their symmetry groups
Halyna Krainichuk
Abstract
The class of all quasigroups is covered by six classes: the class of all asymmetric quasigroups and five varieties of quasigroups (commutative, left symmetric,
right symmetric, semi-symmetric and totally symmetric). Each of these classes
is characterized by symmetry groups of its quasigroups.
In this article, criteria of belonging of group isotopes to each of these classes
are found, including the corollaries for linear, medial and central quasigroups
etc. It is established that an isotope of a noncommutative group is either semisymmetric or asymmetric, each non-medial T-quasigroup is asymmetric etc. The
obtained results are applied for the classification of linear group isotopes of prime
orders, taking into account their up to isomorphism description.
Keywords: central quasigroup, medial quasigroup, isotope, left-, right-,
totally-, semi-symmetric, asymmetric, commutative quasigroup, isomorphism.

Classification: 20N05, 05B15.

Introduction
The group isotope variety is an abundant class of quasigroups in a sense
that it contains quasigroups from almost all quasigroup classes, which
have ever been under consideration. Many authors obtained their results
in the group isotope theory that is why the available results are widely
scattered in many articles and quite often are repeated. However, taking
into account the group theory development, these results have largely a
“folkloric” level of complexity. Considering their applicability, they should
be systematized. One of the attempts of systemic presentations of the
group isotopes is the work by Fedir Sokhatsky “On group isotopes”, which
is given in three articles [17, 18, 19]. But, parastrophic symmetry has
been left unattended.
Some a concept of a symmetry for all parastrophes of quasigroup was
investigated by J.D.H. Smith [16]. This symmetry is known as triality.
The same approach for all parastrophes of quasigroup can be found in such
articles as T. Evans [8], V. Belousov [2], Yu. Movsisyan [14], V. Shcherbacov [24], G. Belyavskaya and T. Popovych [4] and in many articles of
other authors. Somewhat different idea of a symmetry of quasigroups and
loops was suggested by F. Sokhatsky [22]. Using this concept, one can
systematize many results.
The purpose of this article is a classification of group isotopes according
to their parastrophic symmetry groups.
If a σ-parastrophe coincides with a quasigroup itself, then σ is called
a symmetry of the quasigroup. The set of all symmetries forms a group,
which is a subgroup of the symmetry group of order three, i.e., S3 . All
quasigroups defined on the same set are distributed into 6 blocks according
to their symmetry groups. The class of quasigroups whose symmetry

1

contains the given subgroup of the group S3 forms a variety. Thus, there
exist five of such varieties: commutative, left symmetric, right symmetric,
semi-symmetric and totally symmetric. The rest of quasigroups forms the
class of all asymmetric quasigroups, which consists of quasigroups with a
unitary symmetry group.
In this article, the following results are obtained:
• criteria of belonging of group isotopes to each of these classes (Theorem 8);
• a classification of group isotopes defined on the same set, according
to their symmetry groups (Table 1 and Corollary 16);
• corollaries for the well-known classes, such as linear (Corollary 16),
medial (Corollary 10) and central quasigroups (Corollary 13);
• every non-medial T-quasigroup is asymmetric (Corollary 14);
• an isotope of a noncommutative group is either semi-symmetric or
asymmetric (Corollary 9);
• classification of linear isotopes of finite cyclic groups (Corollary 16);
• classification of linear group isotopes of prime orders (Theorem 21).

1

Preliminaries

A groupoid (Q; ·) is called a quasigroup, if for all a, b ∈ Q every of the
equations x · a = b and a · y = b has a unique solution. For every σ ∈ S3
σ
a σ-parastrophe ( · ) is defined by
σ

x1σ · x2σ = x3σ ⇐⇒ x1 · x2 = x3 ,
where S3 := {ι, s, ℓ, r, sℓ, sr} is the symmetric group of order 3 and s :=
(12), ℓ := (13), r := (23).
σ
A mapping (σ; (·)) 7→ ( · ) is an action on the set ∆ of all quasigroup
operations defined on Q. A stabilizer Sym(·) is called a symmetry group
of (·). Thus, the number of different parastrophes of a quasigroup
operation (·) depends on its symmetry group Sym(·). Since Sym(·) is
a subgroup of the symmetric group S3 , then there are six classes of
quasigroups. A quasigroup is called
• asymmetric, if Sym(·) = {ι}, i.e., all parastrophes are pairwise different;
• commutative, if Sym(·) ⊇ {ι, s}, i.e., the class of all commutative
quasigroups is described by xy = yx it means that
s

(·) = ( ·),

ℓ

sr

(·) = ( · ),

r

sℓ

( ·) = ( · );

• left symmetric, if Sym(·) ⊇ {ι, r}, i.e., the class of all left symmetric
quasigroups is described by x · xy = y, it means that
r

(·) = ( ·),

s

ℓ

( ·) = (·),

sℓ

sr

( · ) = ( · );

• right symmetric, if Sym(·) ⊇ {ι, ℓ}, i.e., the class of all right
symmetric quasigroups is described by xy · y = x, it means that
ℓ

(·) = (·),

s

r

( ·) = ( ·),

2

sr

sℓ

( · ) = ( · );

• semi-symmetric, if Sym(·) ⊇ A3 , i.e., the class of all semi-symmetric
quasigroups is described by x · yx = y, it means that
sℓ

sr

(·) = ( · ) = ( · ),

s

ℓ

r

( ·) = (·) = ( ·);

• totally symmetric, if Sym(·) = S3 , i.e., the class of all totally symmetric quasigroups is described by xy = yx and xy · y = x, it means
that all parastrophes coincide.
We will say, a quasigroup has the property of a symmetry, if it satisfies
one of the following symmetry properties: commutativity, left symmetry,
right symmetry, semi-symmetry or total symmetry.
Let P be an arbitrary proposition in a class of quasigroups A. The
proposition σP is said to be a σ-parastrophe of P , if it can be obtained
τ

τ σ −1

from P by replacing every parastrophe ( ·) with ( ·
class of all σ-parastrophes of quasigroups from A.

); σA denotes the

Theorem 1. [22] Let A be a class of quasigroups, then a proposition P
is true in A if and only if σP is true in σA.
Corollary 2. [22] Let P be true in a class of quasigroups A, then σP is
true in σA for all σ ∈ Sym(A).
Corollary 3. [22] Let P be true in a totally symmetric class A, then σP
is true in A for all σ.
A groupoid (Q; ·) is called an isotope of a groupoid (Q; +) iff there
exists a triple of bijections (α, β, γ), which is called an isotopism, such
that the relation x · y := γ −1 (αx + βy) holds. An isotope of a group is
called a group isotope.
A permutation α of a set Q is called unitary of a group (Q; +), if
α(0) = 0, where 0 is a neutral element of (Q; +).
Definition 1. [18] Let (Q; ·) be a group isotope and 0 be an arbitrary
element of Q, then the right part of the formula
x · y = αx + a + βy

(1)

is called a 0-canonical decomposition, if (Q; +) is a group, 0 is its neutral
element and α, β are unitary permutations of (Q; +).
In this case, we say: the element 0 defines the canonical decomposition;
(Q; +) is its decomposition group; α, β are its coefficients and a is its free
member.
Theorem 4. [18] An arbitrary element of a group isotope uniquely defines
a canonical decomposition of the isotope.
Corollary 5. [17] If a group isotope (Q; ·) satisfies an identity
w1 (x) · w2 (y) = w3 (y) · w4 (x)
and the variables x, y are quadratic, then (Q; ·) is isotopic to a commutative group.
Recall, that a variable is quadratic in an identity, if it has exactly
two appearances in this identity. An identity is called quadratic, if all
variables are quadratic. If a quasigroup (Q; ·) is isotopic to a parastrophe
of a quasigroup (Q; ◦), then (Q; ·) and (Q; ◦) are called isostrophic.
The given below Theorem 6 and its Corollary 7 are well known and
can be found in many articles, for example, in [2], [18].

3

Theorem 6. A triple (α, β, γ) of permutations of a set Q is an autotopism
of a group (Q, +) if and only if there exists an automorphism θ of (Q, +)
and elements b, c ∈ Q such that
α = Lc Rb−1 θ,

β = Lb θ,

γ = Lc θ.

Corollary 7. Let α, β1 , β2 , β3 , β4 be permutations of a set Q besides α
is a unitary transformation of a group (Q, +) and let
α(β1 x + β2 y) = β3 u + β4 v,
where {x, y} = {u, v} holds for all x, y ∈ Q. Then the following statements
are true:
1) α is an automorphism of (Q, +), if u = x, v = y;
2) α is an anti-automorphism of (Q, +), if u = y, v = x.
A quasigroup is a linear group isotope [1], if there exists a group (Q; +),
its automorphisms ϕ, ψ, an arbitrary element c such that for all x, y ∈ Q
x · y = ϕx + c + ψy.
T. Kepka, P. Nemec [11, 12] introduced the concept of T -quasigroups
and studied their properties, namely the class of all T -quasigroups is a variety. T -quasigroups, sometimes, are called central quasigroups. Central
quasigroups are precisely the abelian quasigroups in the sense of universal
algebra [26]. A T -quasigroup is a linear isotope of an abelian group and,
according to Toyoda-Bruck theorem [27, 5], it is medial if and only if coefficients of its canonical decompositions commute. A medial quasigroup [2]
is a quasigroup defined by the identity of mediality
xy · uv = xu · yv.

2

Isotopes of groups

In this section, criteria for group isotopes to have symmetry properties are given and a classification of group isotopes according to their
symmetry groups is described.

2.1

Classification of group isotopes

The criteria for group isotopes to be commutative, left symmetric and
right symmetric are found by O. Kirnasovsky [10]. The criteria of total
symmetry, semi-symmetry and asymmetry are announced by the author
in [13].
The following theorem systematizes all criteria on symmetry and implies a classification of group isotopes according to their symmetry groups.
Theorem 8. Let (Q; ·) be a group isotope and (1) be its canonical decomposition, then
1) (Q; ·) is commutative if and only if (Q; +) is abelian and β = α;
2) (Q; ·) is left symmetric if and only if (Q; +) is abelian and β = −ι;
3) (Q; ·) is right symmetric if and only if (Q; +) is abelian and α = −ι;
4) (Q; ·) is totally symmetric if and only if (Q; +) is abelian and α =
β = −ι;

4

5) (Q; ·) is semi-symmetric if and only if α is an anti-automorphism of
(Q; +),
β = α−1 , α3 = −Ia−1 , αa = −a, where Ia (x) := −a + x + a;
6) (Q; ·) is asymmetric if and only if (Q; +) is not abelian or −ι 6= α 6=
β 6= −ι and at least one of the following conditions is true: α is not
an anti-automorphism, β 6= α−1 , α3 6= −Ia−1 , αa 6= −a.
Proof. 1) Let a group isotope (Q; ·) be commutative, e.i., the identity
xy = yx holds. Using its canonical decomposition (1), we have
αx + a + βy = αy + a + βx.
Corollary 5 implies that (Q; +) is an abelian group. When x = 0, we
obtain α = β.
Conversely, let (Q; +) be an abelian group and β = α, then
x · y = αx + a + αy = αy + a + αx = y · x.
Thus, (Q; ·) is a commutative quasigroup.
2) Let a group isotope (Q; ·) be left symmetric, e.i., the identity xy·y =
x holds. Using its canonical decomposition (1), we have
α(αx + a + βy) + a + βy = x.
Replacing a + βy with y, we obtain α(αx + y) = x − y. Corollary 7 implies
that α is an automorphism. When x = 0, we have αy = −y, i.e., α = −ι.
Since α is an automorphism and an anti-automorphism, then (Q; +) is an
abelian group.
Conversely, suppose that the conditions of 2) are performed, then (Q; ·)
is a left symmetric quasigroup. Indeed,
xy · y = −(−x + βy) + βy = x − βy + βy = x.
The proof of 3) is similar to 2). The point 4) follows from 2) and 3).
5) Let a group isotope (Q; ·) be semi-symmetric, e.i., the identity x ·
yx = y holds. Using (1), we have
αx + a + β(αy + a + βx) = y,
hence,
β(αy + a + βx) = −a − αx + y.
Corollary 7 implies that β is an anti-automorphism of (Q; +), therefore
β 2 x + βa + βαy = −a − αx + y.

(2)

When x = y = 0, we obtain βa = −a and, when x = 0, we have βα = ι,
i.e., β = α−1 . Substitute the obtained relations in (2):
α−2 x − a + y = −a − αx + y.
Reducing y on the right in the equality and replacing x with α2 x, we have
x − a = −a − α3 x, wherefrom −α3 x = a + x − a that is α3 = −Ia−1 .
Conversely, suppose that the conditions of 5) hold. A quasigroup (Q; ·)
defined by x · y := αx + a + α−1 y is semi-symmetric. Indeed,
x · yx = αx + a + α−1 (αy + a + α−1 x).

5

Since α is an anti-automorphism of the (Q; ·), then
x · yx = αx + a + α−2 x + α−1 a + y.
α3 = −Ia−1 implies a + α−2 x = −αx + a, so,
x · yx = αx − αx + a + α−1 a + y = a + α−1 a + y.
Because α−1 a = −a, then x · yx = y. Thus, (Q; ·) is semi-symmetric.
6) Asymmetricity of a group isotope means that it is neither commutative, nor left symmetric, nor right symmetric, nor semi-symmetric. It
means that all conditions 1)–4) are false. Falsity of 1) implies falsity of 4).
Falsity of 5) is equivalent to fulfillment of at least one of the conditions:
α is not an anti-automorphism, β 6= α−1 , α3 6= −Ia−1 , α(a) 6= −a. Falsity
of 1), 2), 3) means that (Q; +) is noncommutative or each of the following
inequalities β 6= α, β 6= −ι, α 6= −ι is true. Thus, 6) has been proved.
From Theorem 8, we can deduce the corollary for the classification of
group isotopes over a noncommutative group.
Corollary 9. An isotope of a noncommutative group is either semisymmetric or asymmetric.
Proof. Theorem 8 implies that an isotope of a noncommutative group can
be semi-symmetric or asymmetric. A group isotope can not be asymmetric and semi-symmetric simultaneously, because according to definition,
a symmetry group of a semi-symmetric quasigroup is A3 or S3 and a
symmetry group of an asymmetric quasigroup is {ι}.
Corollary 10. Commutative, left symmetric, right symmetric and totally
symmetric linear isotopes of a group are medial quasigroups.
Proof. The corollary follows from Theorem 8, because in every of these
cases the decomposition group of a canonical decomposition of a group
isotope is commutative and both of its coefficients are automorphisms
according to assumptions and they commute.
Corollary 11. A nonmedial linear isotope of an arbitrary group is either
semi-symmetric or asymmetric.
Proof. The corollary immediately follows from Theorem 8 and Corollary 10.

2.2

Classification of isotopes of abelian groups

An isotope of a nonabelian group is either semi-symmetric or asymmetric
(see Corollary 9). In other words, commutative, left symmetric, right
symmetric and totally symmetric group isotopes exist only among isotopes
of a commutative group. Consequently, it is advisable to formulate a
corollary about classification of isotopes of commutative groups.
Corollary 12. Let (Q; ·) be an isotope of a commutative group and (1) be
its canonical decomposition, then the following conditions are true: 1)-4)
of Theorem 8 and
5′ ) (Q; ·) is semi-symmetric if and only if α is an automorphism of
(Q; +), β = α−1 , α3 = −ι, αa = −a;

6

6′ ) (Q; ·) is asymmetric if and only if −ι 6= α 6= β 6= −ι and at least
one of the following conditions is true: α is not an automorphism,
β 6= α−1 , α3 6= −ι, αa 6= −a.
Proof. The proof follows from Theorem 8 taking into account that (Q; +)
is commutative.
The varieties of medial and T -quasigroups are very important and investigated subclasses of the variety of all group isotopes. These quasigroups have different names in the scientific literature. For example, medial quasigroups also are called entropic or bisymmetry, and T quasigroups are called central quasigroups [26], linear isotopes of commutative groups and etc.
The next statement gives a classification of varieties of T -quasigroups
according to their symmetry groups.
Corollary 13. Let (Q; ·) be a linear isotope of a commutative group and
(1) be its canonical decomposition, then the following conditions are true:
1)-4) of Theorem 8, 5′ ) of Corollary 12 and
6′′ ) (Q; ·) is asymmetric if and only if −ι 6= α 6= β 6= −ι and at least one
of the following conditions is true: β 6= α−1 , α3 6= −ι, αa 6= −a.
Proof. This theorem immediately follows from Corollary 12.
Corollary 14. Every nonmedial T-quasigroup is asymmetric.
In other words, if a T-quasigroup is not asymmetric, then it is medial.

3

Linear isotopes of finite cyclic groups

A full description of all n-ary linear isotopes of cyclic groups up to an isomorphism is given by F. Sokhatsky and P. Syvakivskyj [20]. All pairwise
non-isomorphic group isotopes up to order 15 and a criterion of their existence are established by O. Kirnasovsky [10]. Some algebraic properties
of non-isomorphic quasigroups are studied by L. Chiriac, N. Bobeica and
D. Pavel [6] using the computer.
Let Q be a set and Is(+) be a set of all isotopes of a group (Q; +).
Theorem 8 does not give a partition of Is(+). But it is easy to see that only
totally symmetric quasigroups are common for two classes of symmetric
quasigroups, i.e., of group isotopes being not asymmetric.
To emphasize that a group isotope is not totally symmetric we add
the word ‘strictly’. For example, the term ‘a strictly commutative group
isotopes’ means that it is commutative, but not totally symmetric.
Theorem 8 implies that the conditions of exclusion of totally symmetric
quasigroups from a set of group isotopes are the following: coefficients of
its canonical decomposition do not equal −ι simultaneously. Generally
speaking, a set Is(+) is parted into six subsets.

7

A group isotope
(Q; ·)
is strictly
commutative
is strictly left
symmetric

its symmetry group

conditions of its
canonical decomposition (1)

{ι, s}

(Q; +) is abelian, β = α 6= −ι

{ι, r}

(Q; +) is abelian, β = −ι 6= α

{ι, ℓ}

(Q; +) is abelian, α = −ι 6= β

is strictly right
symmetric
is strictly
semi-symmetric

A3

is totally
symmetric

α is an anti-automorphism of
(Q; +), β = α−1 , αa = −a, α3 =
−Ia−1 , where Ia (x) := −a + x + a,
(Q; +) is non-abelian or α 6= −ι

S3

(Q; +) is abelian α = β = −ι

α is not an anti-automorphism of
(Q; +), β 6= α−1 , α3 6= −Ia−1 ,
is asymmetric
{ι}
αa 6= −a and (Q; +) is non-abelian
or −ι 6= α 6= β 6= −ι.
Table 1. A partition of group isotopes.
Consider the set of all linear isotopes of a finite cyclic group. Their
up to isomorphism description has been found by F. Sokhatsky and
P. Syvakivsky [20].
We will use the following notation: Zm denotes the ring of integers
modulo m; Z∗m the group of invertible elements of the ring Zm ; and
(α, β, d), where α, β ∈ Z∗m and d ∈ Zm , denotes an operation (◦) which is
defined on Zm by the equality
x ◦ y = α · x + β · y + d.

(3)

Since every automorphism θ of the cyclic group (Zm ; +) can be defined by
θ(x) = k · x for some k ∈ Z∗m , then linear isotopes of (Zm ; +) are exactly
the operations being defined by (3), i.e., they are the triples (α, β, d).
Theorem 15. [20] An arbitrary linear isotope of a cyclic m order group
is isomorphic to exactly one isotope (Zm , ◦) defined by (3), where α, β is
a pair of invertible elements in the ring Zm and d is a common divisor of
µ = α + β − 1 and m.
Classification of linear group isotopes of Zm according to their symmetry groups is given in the following corollary.
Corollary 16. Let Zm be a ring of integers modulo m and let (α, β, d)
be its arbitrary linear isotope, where d ∈ GCD(m; α + β − 1). Then an
arbitrary linear isotope of an m-order cyclic group is isomorphic to exactly
one isotope (Zm , ◦) defined by (3), besides
conditions of its canonical decomposition (1)
β = α 6= −1
β = −1 6= α
α = −1 6= β

is strictly commutative
is strictly left symmetric
is strictly right symmetric

its symmetry group
{1, s}
{1, r}
{1, ℓ}

is strictly semi-symmetric

A3

α 6= −1, β = α−1 ,
α3 = −1, αd = −d

is totally symmetric

S3

α = β = −1

is asymmetric

{1}

−1 6= α 6= β 6= −1 and
β 6= α−1 or α3 6= −1 or
αd 6= −d

a group isotope (Q; ◦)

8

3.1

Linear group isotopes of prime orders

Group isotopes and linear isotopes were studied by many authors: V. Belousov [1], E. Falconer [9], T. Kepka and P. Nemec [11], [12] , V. Shcherbacov [23], F. Sokhatsky [21], A. Drápal [7], G. Belyavskaya [3] and others.
In this part of the article, we are giving a full classification of linear
group isotopes of prime order up to isomorphism relation and according
to their symmetry groups.
Theorem 15 implies the following statements.
Corollary 17. [20] Linear group isotopes of a prime order, which are
defined by a pair (α, β) are pairwise isomorphic, if α+β 6= 1. If α+β = 1,
then they are isomorphic to either (α, β, 0) or (α, β, 1).
Corollary 18. [20, 25] There exist exactly p2 −p−1 linear group isotopes
of a prime order p up to isomorphism.
Let p be prime, then Zp is a field, so, according to Corollary 17, there
are two kinds of group isotopes of the cyclic group Zp :
• M0 := {(α, β, 0) | α, β = 1, 2, . . . , p − 1};
• M1 := {(α, 1 − α, 1) | α = 2, . . . , p − 1}.
For brevity, the symbols cs, ls, rs, ts, ss, as denote respectively strictly
commutative, strictly left symmetric, strictly right symmetric, strictly
semi-symmetric, totally symmetric, asymmetric quasigroups. For example, M0ss denotes the set of all strictly semi-symmetric group isotopes from
M0 .

Quasigroups of orders 2 and 3. Every quasigroup of the orders 2
and 3 is isotopic to the cyclic groups. Only ι is a unitary substitutions of
Z2 , so Theorem 4 implies that there exist two group isotopes:
x ◦ y := x + y,
0

and

x ◦ y := x + y + 1.
1

They are isomorphic and ϕ(x) := x + 1 is the corresponding isomorphism.
Proposition 19. All quasigroups of the order 2 are pairwise isomorphic.
There exist two unitary substitutions of the group Z3 : ι and (12)
and the both of them are automorphisms of the cyclic group Z3 . Thus,
Theorem 4 implies that all quasigroups of the order 3 are linear isotopes
of the cyclic group. So, Corollary 17 implies that
M0 = {(1, 1, 0), (2, 2, 0), (1, 2, 0), (2, 1, 0)},

M1 = {(2, 2, 1)}.

According to Corollary 18 and Corollary 16, we obtain the following result.
Proposition 20. There exist exactly five 3-order quasigroups up to isomorphism, which can be distributed into four blocks:
1) strictly commutative: (1, 1, 0);
2) strictly left symmetric: (1, 2, 0);
3) strictly right symmetric: (2, 1, 0);
4) totally symmetric: (2, 2, 0), (2, 2, 1).

9

Linear group isotopes of the order p > 3. Full description of
these group isotopes is given in the following theorem.
Theorem 21. The set of all pairwise non-isomorphic group isotopes of
prime order p (p > 3) is equal to
M = {(α, β, 0) | α, β = 1, 2, . . . , p−1} ∪ {(α, 1−α, 1) | α = 2, . . . , p−1}.
The set M equals union of the following disjoint sets:
1) the set of all strictly commutative group isotopes
M cs = {(1, 1, 0), (2, 2, 0), . . . , (p − 2, p − 2, 0), (2−1 , 2−1 , 1)};
2) the set of all strictly left symmetric group isotopes
M ls = {(1, p − 1, 0), (2, p − 1, 0), . . . , (p − 2, p − 1, 0), (2, p − 1, 1)};
3) the set of all strictly right symmetric group isotopes
M rs = {(p − 1, 1, 0), (p − 1, 2, 0), . . . , (p − 1, p − 2, 0), (p − 1, 2, 1)};
4) the set of all totally symmetric group isotopes
M ts = {(p − 1, p − 1, 0)};
5) the set M ss of all strictly semi-symmetric group isotopes is empty,
if p − 3 is not quadratic residue modulo p, but if there exists k such
that p − 3 = k2 modulo p, then the set is equal to



M ss = (1 + k)2−1 , 2(1 + k)−1 , 0 , (1 − k)2−1 , 2(1 − k)−1 , 0 ;
6) the set of all asymmetric group isotopes is equal to

M as = {(3, p−2, 1), . . . , (p−2, 3−p, 1)} \ (2−1 , 1−2−1 , 1) ∪

∪ (α, β, 0) α, β = 1, 2, 3, . . . , p − 2, α 6= β \ M ss .

Proof. Let (α, β, d) be an arbitrary group isotope.
If it is commutative, then, according to Corollary 16, we obtain
M0cs = {(α, α, 0) | α = 1, 2, . . . , p − 2},

|M0cs | = p − 2.

If d = 1, then Corollary 17 implies 2α = 1, i.e., M1cs = {(2−1 , 2−1 , 1)}.
Thus, the set of all pairwise non-isomorphic commutative group isotopes
is M cs = M0cs ∪ M1cs and |M cs | = p − 1.
Consider left symmetric quasigroups. According to Corollary 16, we
have
M0ls = {(α, p − 1, 0) | α = 1, 2, . . . , p − 2},

|M0ls | = p − 2.

If d = 1, then Corollary 17 implies α = 2 and, by Corollary 16, M1ls =
{(2, −1, 1)}. Thus, the set of all pairwise non-isomorphic left symmetric
group isotopes is M ls = M0ls ∪ M1ls and |M ls | = p − 1.
The relationships for right symmetric quasigroups can be proved in
the same way.
In virtue of Corollary 16, an arbitrary totally symmetric isotope is
defined by the pair (−1, −1) = (p − 1, p − 1) of automorphisms. According
to Corollary 17, d = 0, 1. Suppose that d = 1, then p − 1 + p − 1 = 1, i.e.,

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2p = 3. Since p > 3, than this equality is impossible, so, d = 0. Thus,
M ts = {(p − 1, p − 1, 0)} and |M ts | = 1.
Consider semi-symmetric quasigroups. According to Corollary 17,
α 6= −1,

α3 = −1,

αd = −d,

where d = 0, 1. But d 6= 1, since α 6= −1, so d = 0. The equality α3 = −1
is equivalent to (α + 1)(α2 − α + 1) = 0. It is equivalent to α2 − α + 1 = 0.
It is easy to prove that α exists if and only if p − 3 is a quadratic residue
modulo p. If p − 3 = k2 modulo p, then



M ss = (1 + k)2−1 , 2(1 + k)−1 , 0 , (1 − k)2−1 , 2(1 − k)−1 , 0 .
Consequently, |M ss | = 2, and M ss = ∅ otherwise.
Let (α, β, d) be an arbitrary asymmetric group isotope. Corollary 17
implies that d ∈ {0, 1}. Since the given isotope is neither commutative,
nor left symmetric, nor right symmetric, nor totally symmetric, then,
according to Corollary 16, α, β 6∈ {0, p − 1} and α 6= β.
In the case when d = 1, then from Corollary 17, it follows that β = 1−α
and the relationships α, β 6∈ {0, p − 1}, α 6= β imply
α 6∈ {0, 1, 2,

p+1
, p − 1}.
2

Since for d = 1 semi-symmetric group isotopes do not exist, then


p+1
,
M1as = (α, 1 − α, 1) α = 3, . . . , p − 2, α 6=
2
|M1as | = p − 5.
The set M0as depends on existence of semi-symmetric isotopes. Nevertheless,

M0as = (α, β, 0) α, β = 1, 2, 3, . . . , p − 2, α 6= β \ M ss ,
 2
p − 5p + 4, if p − 3 is a quadratic residue modulo p;
|M0as | =
p2 − 5p + 6, otherwise.
Note, that F. Radó [15] proved that a semi-symmetric group isotope
of prime order p exists if and only if p − 3 is a quadratic residue modulo
p.
The general formula of all linear pairwise non-isomorphic group isotopes of the prime order p is found in Corollary 18 by F. Sokhatsky and
P. Syvakivskyj [20] and also by K. Shchukin [25].
Corollary 22. A number of all linear group isotopes of the prime order
p > 3 up to isomorphism is equal to p2 − p − 1 and it is equal to the sum
of the following numbers:
1) p − 1 of strictly commutative quasigroups;
2) p − 1 of strictly left symmetric quasigroups;
3) p − 1 of strictly right symmetric quasigroups;
4) 1 of the totally symmetric quasigroup;
5) 2 of semi-symmetric quasigroups, if p−3 is quadratic residue modulo
p and 0 otherwise;

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6) (p − 2)2 − 5 asymmetric quasigroups, if p − 3 is quadratic residue
modulo p and (p − 2)2 − 3 otherwise.
Proof. The proof immediately follows from the proof of Theorem 21.
Acknowledgment. The author is grateful to her scientific supervisor
Prof. Fedir Sokhatsky for the design idea and the discussion of this article,
to the members of his scientific School for helpful discussions and to the
reviewer of English Vira Obshanska.

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Department of mathematical analysis and differential equations,
Faculty of Mathematics and Information Technology
Donetsk National University
Vinnytska oblast, Ukraine 21000
e-mail: kraynichuk@ukr.net

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