On endomorphisms of groups of orders 37 – 47

It is proved that the finite groups of orders 37–47 are determined by their endomorphism monoids in the class of all groups.


INTRODUCTION
Let G be a group.If for each group H such that the monoids End(G) and End(H) are isomorphic implies an isomorphism between G and H, we say that the group G is determined by its endomorphism monoid in the class of all groups.Examples of such groups are: finite Abelian groups ( [12], Theorem 4.2), generalized quaternion groups ( [13], Corollary 1), torsion-free divisible Abelian groups ( [16], Theorem 1), etc.
The endomorphisms of groups have gained much attention in the past few years due to their applications in generalized linear finite dynamical systems (in this case, the finite vector space and a polynomial map f are replaced with a group and its endomorphism, respectively) [3].Also, Grigorchuk and Mamaghani [7] used iterations of an endomorphism of a group for constructing the groups with prescribed properties such as to have intermediate growth or to be amenable.Different authors have studied so-called E-groups (recall that a group G is called an E-group if its each element commutes with all of images under endomorphisms of G).
The question of which groups are determined by their endomorphism monoids in the class of all groups, and how to find all non-isomorphic groups with isomorphic endomorphism monoids, is one of the main questions in the group theory.For example, in 2014 the following question was asked in the internet forum math.stackexchange:"Can finite non-isomorphic groups of the same order have isomorphic endomorphism monoids?" [4].
In a number of our papers we have made efforts to describe some classes of finite groups which are determined by their endomorphism monoids in the class of all groups.
Computer algebra software GAP [24] provides access to small groups1 such as the presentation of a group and structure description of a group.The software GAP is becoming a useful tool for mathematicians working in the group theory.For example, Chu et al. [6] used the GAP for solving Noether's problem for groups of order 243.Combining the software GAP with the results obtained in [12][13][14]18,19] it is possible to obtain the list of groups of given order n that are determined by their endomorphism monoids.However, the groups of order n not listed there have to be studied using techniques different from the ones used in [12][13][14]18,19] or ad-hoc techniques.Recall that the finite direct product of groups is determined by its endomorphism monoid if all these direct factors are determined by their endomorphism monoids [12,Theorem 1.13].Therefore, if a group G is the direct product of groups G 1 , . . ., G n , and each of them are determined by their endomorphism monoids, then so is G.But if a direct factor, say G i , is not determined by its endomorphism monoid, i.e.End(G i ) ∼ = End(H) for some group have isomorphic endomorphism monoids.For example, it was shown in [20] that the alternating group A 4 and the binary tetrahedral group B ∼ = SL 2 (GF(3)) have isomorphic endomorphism monoids.Studying the determinability of groups of order 36, we proved that the direct products C 3 × A 4 and C 3 × B have isomorphic endomorphism monoids too [9,Theorem 6.1].On the other hand, if a finite group G is a direct product of all pairwise non-isomorphic groups with isomorphic endomorphism monoids, then the group G is determined by End(G) [15].Therefore, the direct product A 4 × B is determined by its endomorphism monoid, but both direct factors are not.In what follows we discuss a simple2 (and naive) method for testing whether a given direct product of finite groups is determined by its endomorphism monoid or not: either all direct factors are determined by their endomorphism monoids, respectively, or the set of direct factors includes a complete set of pairwise non-isomorphic groups with isomorphic endomorphism monoids.This motivates us to study which small groups are determined by their endomorphism monoids.Furthermore, since a similar technique can be applied to several groups of different order, we are also motivated to use a computer 3 and development of necessary algorithms.The results of this paper can be used in computer programs about finite groups and their applications.
We know a complete answer to this problem for finite groups of order less than 37.Among the groups of order less than 37 there are only five groups that are not determined by their endomorphism monoids in the class of all groups: • the alternating group A 4 (also called the tetrahedral group); the cyclic group of order 3 and the alternating group A 4 ; The alternating group A 4 (also called the tetrahedral group) and the binary tetrahedral group B = ⟨a, b | b 3 = 1, aba = bab⟩ are the only groups of order less than 36 that are not determined by their endomorphism monoids in the class of all groups [9,[20][21][22][23] (for some groups of order 32 the proofs are under publishing).In [9], it is proved that among groups of order 36 only the groups C 3 × A 4 , C 1 , and C 2 are not determined by their endomorphism monoids in the class of all groups.Namely, the following results were proved in [9]: (this group is isomorphic to a semidirect product Q C 9 of the quaternion group Q and the cyclic group C 9 of order 9).• The endomorphism monoid of a group G is isomorphic to the endomorphism monoid of C 2 if and only if G = C 2 or G is isomorphic to the following group of order 108: In this paper, we give the solution to the problem for the groups of orders 37-47.We prove the following theorem: Theorem 1.1 (Main theorem).The finite groups of orders 37-47 are determined by their endomorphism monoids in the class of all groups.
The proof of the theorem follows from Theorems 3.1, 4.1, 5.2, and 6.2.We shall use the following notations: the cyclic group of order k; A 4 -the alternating group of order 12 (the tetrahedral group); Z k -the residue class ring Z/kZ; ⟨K, . . ., g, . ..⟩ -the subgroup generated by subsets K, . . .and elements g, . ..; g -the inner automorphism of G, generated by an element g ∈ G; I(G) -the set of all idempotents of End(G); The sets K(x), V (x), H(x), P(x), and J(x) are submonoids of End(G), furthermore, V (x) is a subgroup of Aut(G).We shall write the mapping right from the element on which it acts.

PRELIMINARIES
For the convenience of the reader, let us recall some known facts that will be used in the proofs of our main results.
and K(x) is a submonoid with the identity element x of End(G) which is canonically isomorphic to End(Im x).Under this isomorphism element y of K(x) corresponds to its restriction onto the subgroup Im x of G.
We omit the proofs of these lemmas, because these are straightforward corollaries from the definitions.

Lemma 2.9 ([12], Theorem 1.13). If G and H are groups such that their endomorphism monoids are isomorphic and G splits into a direct product G
From here follows Lemma 2.10.Lemma 2.10.If groups G 1 and G 2 are determined by their endomorphism monoids in the class of all groups, then so is their direct product G 1 × G 2 .

Lemma 2.11 ([12], Theorem 4.2). Every finite Abelian group is determined by its endomorphism monoid in the class of all groups.
Lemma 2.12 ([14], Theorem).Each finite symmetric group is determined by its endomorphism monoid in the class of all groups.

Lemma 2.13 ([19], Section 5). The dihedral group D n is determined by its endomorphism monoid in the class of all groups.
Lemma 2.14 ([13], Corollary 1).The quaternion group Q is determined by its endomorphism monoid in the class of all groups.Lemma 2.15 ([18], Theorem).A semidirect product G = C p n C m , where p is a prime, and n and m are some positive integers, is determined by its endomorphism monoid in the class of all groups.Lemma 2. 16.
) satisfies 1 0 and 2 0 , then there exist a, b, c ∈ D 8 such that where

Denote by x and x i the projections of G onto K and G
) ) (2.6) and for each i, j ∈ { 1, 2, . . ., n }, i ̸ = j, there exists z i j = z ji ∈ I(G) which satisfies the following properties: Conversely, suppose that there exist idempotents x, x 1 , . . ., x n of End(G) such that (2.7) holds and for each i, j ∈ { 1, 2, . . ., n }, i ̸ = j, there exists z i j = z ji ∈ I(G) which satisfies properties 1 0 and 2 0 .Then the group G decomposes into the semidirect product (2.2),where equalities (2.3)- (2.6)  Denote by C (x; x 1 , . . ., x n ) the set of the conditions for x; x 1 , . . ., x n given in the second part of Lemma 2.17 (i.e., equalities (2.7) and 1 0 , 2 0 ).Suppose that the condition C (x; x 1 , . . ., x n ) is satisfied and denote by π C the projection of G onto its subgroup

GROUPS OF ORDERS 37-39 AND 41-47
The group theoretical computer algebra system GAP provides access to descriptions of small order groups [12].Following [12], the groups of orders 37-39 and 41-47 are: In view of Lemmas 2.11, 2.12, 2.13, 2.10, and 2.15, the groups G 1 -G 7 and G 9 -G 22 are determined by their endomorphism monoids in the class of all groups.
Let us consider the group The group G 8 can be presented as follows: By Lemma 2.15, the group G 8 is determined by its endomorphism monoid in the class of all groups.We have proved Theorem 3.1.The finite groups of orders 37-39 and 41-47 are determined by their endomorphism monoids in the class of all groups.

GROUPS OF ORDER 40
According to [24], there exist 14 pairwise non-isomorphic groups of order 40: Lemmas 2.11, 2.13, 2.10, 2.15, and 2.14 imply the following theorem We consider the groups G 6 and G 10 in the next two sections.

GROUP G 6
Let us consider the group The group G 6 can be presented as follows: In this section, we shall prove the following theorem.

Theorem 5.1. A finite group G is isomorphic to G 6 if and only if there exists x ∈ I(G) such that the following properties hold:
Proof.Necessity.Let G = G 6 and G be given by the generating relations as presented above.Denote by x the projection of G onto its subgroup Q = ⟨a, b⟩.Then Im x = ⟨a, b⟩ and Ker x = ⟨c⟩.We have to prove that x satisfies properties 1 0 -7 0 .for some i, j ∈ Z 5 .The map y given by (5.1) preserves the generating relations of G and can be extended to an endomorphism of G if and only if i ∈ Z 5 and j = 0.Each such endomorphism is an idempotent.Therefore, | [x]| = 5 and property 3 0 holds.By Lemma 2.5, J(x) consists of the endomorphisms y of G, where (5. 2) The map y given by (5.2) preserves the generating relations of G and can be extended to an endomorphism of G if and only if i = 0, i.e., y = 0. Hence property 4 0 is true.The set V (x) consists of the automorphisms y of G such that yx = x, i.e., g −1 • gx ∈ Ker x = ⟨c⟩ for each g ∈ G and ay = ac i , by = bc j , cy = c k ( for some i, j, k ∈ Z 5 .The map y given by (5.3) preserves the generating relations of G and can be extended to an endomorphism of G if and only if j = 0.This endomorphism is an automorphism if and only if k and 5 are coprime.Hence |V (x)| = 4 • 5 and property 5 0 holds.By Lemma 2.7, P(x) consists of the endomorphisms y of G such that ay = a, by = b, cy = c i , i ∈ Z 5 . ( The map y given by (5.4) preserves the generating relations of G and can be extended to an endomorphism of G for each i ∈ Z 5 .It follows from here that the submonoid P(x) of End(G) is isomorphic to End(C 5 ), i. e., x satisfies property 6 0 .It was proved that [x] consists of maps y i , i ∈ Z 5 , where By Lemmas 2.2 and 2.3, In view of [13], Lemma 1 and Theorem 14, the nilpotent elements of K(y i ) consist of maps z jk ( j, k ∈ Z 2 ), where It follows that K 0 (y) = K 0 (x) for each y ∈ [x] and property 7 0 is true.The necessity is proved.
Sufficiency.Let G be a finite group such that there exists x ∈ I(G) which satisfies properties 1 0 -7 0 of the theorem.By property 3 0 , x is non-trivial (x ̸ ∈ {0, 1}).
By Lemma 2.1, G = Ker x Im x. (5.5) Denote M = Ker x.The semidirect product (5.5) is not a direct product, because otherwise the projection of G onto its subgroup Ker x is a non-trivial element in J(x) which contradicts property 4 0 .Lemma 2.3 and property 1 0 imply Since the quaternion group is determined by its endomorphism monoid in the class of all groups (Lemma 2.14), we have Im x ∼ = Q and (we identified Im x and Q).In view of (5.6) and property 5 0 , we have (5.7) It follows that all 5 ′ -elements of M belong into the centre of G and where M 5 and M 5 ′ are a Sylow 5-subgroup and a Hall 5 ′ -subgroup of G, respectively.Hence Denote by π the projection of G onto its subgroup M 5 ′ .Then π ∈ J(x) and, by property 4 0 , π = 0. Therefore, M 5 ′ = ⟨1⟩, M = M 5 , and M = Ker x is a 5-group.By (5.7), M is an Abelian 5-group.Let us consider the map where i is an integer.Since M is Abelian, it is easy to check that y i can be uniquely extended to an endomorphism of G and y i ∈ P(x).Clearly, y i y j = y i• j for each integer i and j.Therefore, property 6 0 implies that h 5 = 1 for each h ∈ M and M is an elementary Abelian 5-group.In view of (5.7), there exist c ∈ M such that We can assume that ac ̸ = ca, because the case bc ̸ = cb can be considered similarly.Therefore, by (5.8), there exist i ∈ Z 5 , i ̸ = 0, and If c 0 ̸ = 1, there exists c 1 ∈ M ∩ Z(G) such that c 0 = c i 1 and we have i.e., we can replace the element c by the element cc 1 .It follows that we can assume that c −1 ac = ac i , i ̸ = 0. (5.9) In view of (5.7) and property 3 0 , [x] = {x c j | j ∈ Z 5 }.
Choose j ∈ Z 5 and denote y = x c j .The nilpotent endomorphisms of Q = ⟨a, b⟩ are the maps by Lemma 1 and Theorem 14 in [12].Therefore, the nilpotent endomorphisms of K(y) consist of maps z kl , where (c − j ac j )z kl = c − j a 2k c j , (c − j bc j )z kl = c − j a 2l c j , hz kl = 1, h ∈ M.
Since c ∈ M, we have az kl = c − j a 2k c j , bz kl = c − j a 2l c j .
Theorem 5.2.The group G 6 is determined by its endomorphism monoid in the class of all groups.
Proof.Let G * be a group such that the endomorphism monoids of G * and G 6 are isomorphic: End(G * ) ∼ = End(G 6 ). (5.14) Denote by z * the image of z ∈ End(G 6 ) in isomorphism (5.14).Since End(G * ) is finite, so is G * ([1], Theorem 2).By Theorem 5.1, there exists x ∈ I(G 6 ) which satisfy properties 1 0 − 7 0 of the theorem.In view of isomorphism (5.14), the endomorphism x * satisfies properties 1 0 − 7 0 , where always x and y are replaced by x * and y * , respectively.Using now Theorem 5.1 for G * , it follows that G * and G 6 are isomorphic.The theorem is proved.

ON ENDOMORPHISMS OF G 10
Let us consider the group Our aim is to prove the following theorem.

CONCLUSIONS
We studied the determinability of groups of orders 37-47 by their endomorphism monoids.We proved that all these groups are determined by their endomorphism monoids in the class of all groups.The technique developed in this paper can be applied to other small groups and it can be implemented in the software GAP.
are true.Moreover, the set B = { y ∈ I(G) | x 1 , . . ., x n ∈ K(y) } is non-empty and there exists a unique z ∈ B such that zy = yz = z for each y ∈ B. The endomorphism z is the projection of G onto its subgroup (G 1 ×. ..×G n ) K and Ker z = H.

Theorem 4 . 1 .
The groups G 1 -G 5 , G 7 -G 9 , and G 11 -G 14 are determined by their endomorphism monoids in the class of all groups.

Lemma 2 .
4 and property 2 0 imply that Ker x is a 2 ′ -group, i.e., Im x is a Sylow 2-subgroup of G. Since the Sylow 2-subgroups of G are conjugate, Lemma 2.1 and property 3 0 imply that there exist 5 Sylow 2-subgroups of G and [M : N M (Im x)] = 5. (5.6)
where B is the binary tetrahedral group.•The endomorphism monoid of a group G is isomorphic to the endomorphism monoid of C 1 if and only if G = C 1 or G is isomorphic to the following group of order 72: