ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 17 (2019) 467-479 https://doi.org/10.26493/1855-3974.1892.78f (Also available at http://amc-journal.eu) On separable abelian p-groups Grigory Ryabov * Novosibirsk State University, 1 Pirogova st., 630090, Novosibirsk, Russia, and Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia Received 29 December 2018, accepted 3 April 2019, published online 13 November 2019 An S-ring (a Schur ring) is said to be separable with respect to a class of groups K if every algebraic isomorphism from the S-ring in question to an S-ring over a group from K is induced by a combinatorial isomorphism. A finite group is said to be separable with respect to K if every S-ring over this group is separable with respect to K. We provide a complete classification of abelian p-groups separable with respect to the class of abelian groups. Keywords: Isomorphisms, Schur rings, p-groups. Math. Subj. Class.: 05E30, 05C60, 20B35 1 Introduction Let G be a finite group. A subring of the group ring ZG is called an S-ring (a Schur ring) over G if it is determined in a natural way by a special partition of G (the exact definition is given in Section 2). The classes of the partition are called the basic sets of the S-ring. The concept of the S-ring goes back to Schur and Wielandt. They used S-rings to study a permutation group containing a regular subgroup [19, 20]. For more details on S-rings and their applications we refer the reader to [13]. Let A and A1 be S-rings over groups G and G' respectively. An algebraic isomorphism from A to A' is a ring isomorphism inducing a bijection between the basic sets of A and the basic sets of A'. Another type of an isomorphism of S-rings comes from graph theory. A combinatorial isomorphism from A to A' is defined to be an isomorphism of the corresponding Cayley schemes (see Subsection 2.2). Every combinatorial isomorphism induces the algebraic one. However, the converse statement is not true (the corresponding examples can be found in [6]). *The work is supported by the Russian Foundation for Basic Research (project 17-51-53007). E-mail address: gric2ryabov@gmail.com (Grigory Ryabov) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 468 Ars Math. Contemp. 17(2019)467-479 Let K be a class of groups. Following [3], we say that an S-ring A is separable with respect to K if every algebraic isomorphism from A to an S-ring over a group from K is induced by a combinatorial one. We call a finite group separable with respect to K if every S-ring over G is separable with respect to K (see [18]). The importance of separable S-rings comes from the following observation. Suppose that an S-ring A is separable with respect to K. Then A is determined up to isomorphism in the class of S-rings over groups from K only by the tensor of its structure constants (with respect to the basis of A corresponding to the partition of the underlying group). Given a group G denote the class of groups isomorphic to G by KG. If G is separable with respect to KG then the isomorphism of two Cayley graphs over G can be verified efficiently by using the Weisfeiler-Leman algorithm [12]. In the sense of [10] this means that the Weisfeiler-Leman dimension of the class of Cayley graphs over G is at most 3. More information concerned with separability and the graph isomorphism problem is presented in [3, 17]. Denote the classes of cyclic and abelian groups by KC and KA respectively. The cyclic group of order n is denoted by Cn. In the present paper we are interested in abelian groups and especially in abelian p-groups which are separable with respect to KA. The problem of determining of all groups separable with respect to a given class K seems quite complicated even for K = KC. Examples of cyclic groups which are non-separable with respect to KC were found in [6]. In [5] it was proved that cyclic p-groups are separable with respect to KC. We prove that a similar statement is also true for KA. Theorem 1.1. For every prime p a cyclic p-group is separable with respect to KA. The result obtained in [18] implies that an abelian group of order 4p is separable with respect to KA for every prime p. From [9] it follows that for every group G of order at least 4 the group G x G is non-separable with respect to KGxG. One can check that a normal subgroup of a group separable with respect to KA is separable with respect to Ka (see also Lemma 2.5). The above discussion shows that a non-cyclic abelian p-group separable with respect to KA is isomorphic to Cp x Cpk or Cp x Cp x Cpk, wherep e {2,3} and k > 1. The separability of the groups from the first family was proved in [17]. In the present paper we study the question on the separability of the groups from the second family. Theorem 1.2. The group Cp x Cp x Cpk, where p e {2, 3} and k > 1, is separable with respect to KA if and only if k = 1. As an immediate consequence of Theorem 1.1, Theorem 1.2, and the above mentioned results, we obtain a complete classification of abelian p-groups separable with respect to Ka. Theorem 1.3. An abelian p-group is separable with respect to KA if and only if it is cyclic or isomorphic to one of the following groups: C2 x C2fc, C3 x C%k, C23, C3, where k 1. Throughout the paper we write for short "separable" instead of "separable with respect to Ka". The text is organized in the following way. Section 2 contains a background of G. Ryabov: On separable abelian p-groups 469 S-rings. Section 3 is devoted to S-rings over cyclic p-groups. We finish Section 3 with the proof of Theorem 1.1. In Section 4 we prove Theorem 1.2. The author would like to thank Prof. I. Ponomarenko for the fruitful discussions on the subject matters and Dr. Sven Reichard for the help with computer calculations. Notation. • The ring of rational integers is denoted by Z. • Let X C G. The element J2xex x of the group ring ZG is denoted by X. • The order of g G G is denoted by |g|. • The set {x-1 : x G X} is denoted by X-1. • The subgroup of G generated by X is denoted by (X}; we also set rad(X) = {g G G : gX = Xg = X}. • If m G Z then the set {xm : x G X} is denoted by X(m). • Given a set X C G the set {(g, xg) : x G X, g G G} of edges of the Cayley graph Cay(G, X) is denoted by R(X). • The group of all permutations of a set Q is denoted by Sym(Q). • The subgroup of Sym(G) induced by right multiplications of G is denoted by Gright. • For a set A C Sym(G) and a section S = U/L of G we set AS = {fS : f G A, Sf = S}, where Sf = S means that f permutes the L-cosets in U and fS denotes the bijection of S induced by f. • If a group K acts on a set Q then the set of all orbtis of K on Q is denoted by Orb(K, Q). • If H < G then the normalizer of H in G is denoted by NG(H). • If K < Sym(Q) and a G Q then the stabilizer of a in K is denoted by Ka. • The cyclic group of order n is denoted by Cn. 2 S-rings In this section we give a background of S-rings. The most of definitions and statements presented here are taken from [13, 17]. 2.1 Definitions and basic facts Let G be a finite group and ZG the group ring over the integers. The identity element of G is denoted by e. A subring A C ZG is called an S-ring over G if there exists a partition S = S(A) of G such that: (1) {e} G S, (2) if X G S then X-1 G S, (3) A = SpanZ{X : X G S}. The elements of S are called the basic sets of A and the number |S| is called the rank of A. Given X, Y, Z G S the number of distinct representations of z G Z in the form z = xy with x G X and y G Y is denoted by cX Y. If X and Y are basic sets of A then XY = J2zes(A) cX yZ. So the integers cX Y are structure constants of A with respect 470 Ars Math. Contemp. 17(2019)467-479 to the basis {X : X G S}. It is easy to verify that given basic sets X and Y the set XY is also basic whenever |X| = 1 or |Y| = 1. A set X C G is said to be an A-set if X G A. A subgroup H < G is said to be an A-subgroup if H is an A-set. One can check that for every A-set X the groups (X} and rad(X) are A-subgroups. A section U/L is said to be an .A-section if U and L are A-subgroups. If S = U/L is an A-section then the module AS = SpanZ {Xn : X G S(A), X C U} , where n: U ^ U/L is the canonical epimorphism, is an S-ring over S. If K < Aut(G) then the set Orb(K, G) forms a partition of G that defines an S-ring A over G. In this case A is called cyclotomic and denoted by Cyc(K, G). Let G be abelian. Then from Schur's result [19] it follows that X(m) G S(A) for every X G S(A) and every m coprime to |G|. We say that X, Y G S(A) are rationally conjugate if Y = X(m) for some m coprime to |G|. 2.2 Isomorphisms and schurity Throughout this and the next two subsections A and A' are S-rings over groups G and G' respectively. A bijection f: G ^ G' is called a (combinatorial) isomorphism from A over to A' if {R(X)f : X G S(A)} = {R(X') : X' G S(A')}, where R(X )f = {(gf, hf) : (g, h) G R(X)}. If there exists an isomorphism from A to A' we write A = A'. The group Iso(A) of all isomorphisms from A onto itself has a normal subgroup Aut(A) = {f G Iso(A) : R(X)f = R(X) for every X G S(A)}. This subgroup is called the automorphism group of A. Note that Aut(A) > Gright. If S is an A-section then Aut(A)S < Aut(AS). An S-ring A over G is said to be normal if Gright < Aut(A). One can check that NAut(A) (Gright)e = Aut(A) O Aut(G). (2.1) Now let K be a subgroup of Sym(G) containing Gright. As Schur proved in [19], the Z-submodule V(K, G) = SpanZ{X : X G Orb(Ke, G)}, is an S-ring over G. An S-ring A over G is called schurian if A = V(K, G) for some K such that Gright < K < Sym(G). Not every S-ring is schurian. The first example of a non-schurian S-ring was found by Wielandt in [20, Theorem 25.7]. It is easy to see that A is schurian if and only if S(A) = Orb(Aut(A)e, G). (2.2) Every cyclotomic S-ring is schurian. More precisely, if A = Cyc(K, G) for some K < Aut(G) then A = V(Gright x K, G). G. Ryabov: On separable abelian p-groups 471 2.3 Algebraic isomorphisms and separability A bijection y: S(A) ^ S(A') is called an algebraic isomorphism from A to A' if cx,y = cXv,Y v for all X, Y, Z G S(A). The mapping X ^ Xv is extended by linearity to the ring isomorphism of A and A'. This ring isomorphism we denote also by y. If there exists an algebraic isomorphism from A to A' then we write A =Alg A'. An algebraic isomorphism from A to itself is called an algebraic automorphism of A. The group of all algebraic automorphisms of A is denoted by AutAlg (A). Every isomorphism f of S-rings preserves the structure constants and hence f induces the algebraic isomorphism y f. However, not every algebraic isomorphism is induced by a combinatorial one (see [6]). Let K be a class of groups. An S-ring A is defined to be separable with respect to K if every algebraic isomorphism from A to an S-ring over a group from K is induced by a combinatorial isomorphism. Put AutAlg(A)0 = {y G AutAlg(A) : y = yf for some f G Iso(A)}. It is easy to see that yf = yg for f, g G Iso(A) if and only if gf-1 G Aut(A). Therefore | AutAlg(A)o| = | Iso(A)|/| Aut(A)|. (2.3) One can verify that for every group G the S-ring of rank 2 over G and ZG are separable with respect to the class of all finite groups. In the former case there exists the unique algebraic isomorphism from the S-ring of rank 2 over G to the S-ring of rank 2 over a given group of order |G| and this algebraic isomorphism is induced by every bijection. In the latter case every basic set is singleton and hence every algebraic isomorphism is induced by an isomorphism in a natural way. Let y: A ^ A' be an algebraic isomorphism. One can check that y is extended to a bijection between A- and A'-sets and hence between A- and A'-sections. The images of an A-set X and an A-section S under these extensions are denoted by Xv and Sv respectively. If S is an A-section then y induces the algebraic isomorphism yS: AS ^ A'S,, where S' = Sv. The above bijection between the A- and A'-sets is, in fact, an isomorphism of the corresponding lattices. One can check that (X= (X and rad(Xv) = rad(X )v for every A-set X (see [4, Equation (10)]). Since cX?}Y = SY,X -i |X|, where X, Y G S(A) and ¿Y,X-i is the Kronecker delta, we conclude that |X| = cXe}X-i, (X-1)v = (Xv)-1, and |X| = |Xfor every A-set X. In particular, |G| = |G'|. , 2.4 Cayley isomorphisms A group isomorphism f: G ^ G' is called a Cayley isomorphism from A to A' if S(A)f = S(A'). If there exists a Cayley isomorphism from A to A' we write A =Cay A'. Every Cayley isomorphism is a (combinatorial) isomorphism, however the converse statement is not true. 472 Ars Math. Contemp. 17(2019)467-479 2.5 Algebraic fusions Let A be an S-ring over G and $ < AutAlg(A). Given X G S(A) put X$ = |JX The partition {X$ : X G S(A)} defines an S-ring over G called the algebraic fusion of A with respect to $ and denoted by A$. Suppose that $ = {yf : f G K} for some K < Iso(A) and A is schurian. Then one can verify that A$ = V(Aut(A)K, G). In particular, the following statement holds. Lemma 2.1. Let A be a schurian S-ring over G and K < Iso(A). Then A$, where $ = {yf : f G K}, is also schurian. 2.6 Wreath and tensor products Let A be an S-ring over a group G and S = U/L an A-section. The S-ring A is called the S-wreath product if L < G and L < rad(X) for all basic sets X outside U. In this case we write A = Au Is AG/L. The S-wreath product is called non-trivial or proper if e = L and U = G. If U = L we say that A is the wreath product of AL and AG/L and write A = AL IAG/L. Let Ai and A2 be S-rings over groups Gi and G2 respectively. Then the set S = S(A1) x S(A2) = {X1 x X2 : X1 G S(A1),X2 G S(A2)} forms a partition of G = G1 x G2 that defines an S-ring over G. This S-ring is called the tensor product of A1 and A2 and denoted by A1 ( A2. Lemma 2.2. The tensor product of two separable S-rings is separable. Proof. As noted in [18, Lemma 2.6], the statement of the lemma follows from [1, Theorem 1.20]. □ Lemma 2.3 ([17, Lemma 4.4]). Let A be the S-wreath product over an abelian group G for some A-section S = U/L. Suppose that Au and AG/L are separable and Aut(AU )S = Aut(AS). Then A is separable. In particular, the wreath product of two separable S-rings is separable. Let Q be a finite set. Permutation groups K, K' < Sym(^) are called 2-equivalent if Orb(K, Q2) = Orb(K', Q2). A permutation group K < Sym(Q) is called 2-isolated if it is the only group which is 2-equivalent to K. Lemma 2.4. Let A be the S-wreath product over an abelian group G for some A-section S = U/L. Suppose that Au and AG/L are separable, Au is schurian, and the group Aut(AS) is 2-isolated. Then A is separable. Proof. Since Au is schurian, the groups Aut(AU)S and Aut(AS) are 2-equivalent. Indeed, Orb(Aut(AU)S, S2) = Orb(Aut(AS),S2) = {R(X) : X G S(AS)}. G. Ryabov: On separable abelian p-groups 473 This implies that Aut(A^)S = Aut(AS) because Aut(AS) is 2-isolated. Therefore the conditions of Lemma 2.3 hold and A is separable. □ Lemma 2.5. Let H be a normal subgroup of a group G, B an S-ring over H, ^ G AutAig(B) \ AutAig(B)0. Then there exists ^ G AutAig(A) \ AutAig(A)o, where A = B l Z(G/H), such that ^H = y. Proof. Define ^ as follows: X^ = X^ for X G S(AH) and X^ = X for X G S(A) \ S(Ah). Let us prove that ^ G AutAig (A). To do this it suffices to check that cXv yv — c|,Y for all X, Y, Z G S(A). Suppose that X, Y G S(Ah). If Z G S(Ah) then yv = y v = cXy .If Z G S(Ah ) then Z^ G S(AH) and hence cXV y v = cX y = 0. Now suppose that exactly one of the sets X, Y, say X, lies inside H. Then Y^ = Y and X u X^ C H < rad(Y). So XY = X^Y = |X|Y. This implies that , yv = cX , y = |X| whenever Z = Y and cXV y v = cX , y = 0 otherwise. Finally, suppose that X, Y G S(Ah ). In this case X ^ = X and Y ^ = Y .If Z G S(Ah) then Z^ = Z and hence cfV y v = cX,y. If Z G S(AH) then Z and Z^ enter the element XY with the same coefficients because H = rad(X) n rad(Y). Therefore cX;y v = cX,y. Thus, ^ G AutAig (A). If ^ is induced by an isomorphism then [4, Lemma 3.4] implies that = ^ is also induced by an isomorphism. We obtain a contradiction with the assumption of the lemma and the lemma is proved. □ 3 S-rings over cyclic p-groups In this section we prove Theorem 1.1. Before the proof we recall some results on S-rings over cyclic p-groups. The most of them can be found in [7, 8]. Throughout the section p is an odd prime, G is a cyclic p-group and A is an S-ring over G. We say that X G S(A) is highest if X contains a generator of G. Put rad(A) = rad(X), where X is highest. Note that rad(A) does not depend on the choice of X because every two basic sets are rationally conjugate and hence have the same radicals. Lemma 3.1. The S-ring A is schurian and one of the following statements holds for A: (1) | rad(A)| = 1 and rk(A) = 2; (2) | rad(A)| = 1, A is normal, and A = Cyc(K, G) for some K < K0, where K0 is the subgroup of Aut(G) of order p — 1; (3) | rad(A) | > 1 and A is the proper generalized wreath product. Proof. The S-ring A is schurian by the main result of [16]. The other statements of the lemma follow from [8, Theorem 4.1, Theorem 4.2 (1)] and [7, Lemma 5.1, Equation (1)]. □ Lemma 3.2. Let S be an A-section with |S| > p2. The following statements hold: (!) If Statement (2) of Lemma 3.1 holds for A then Statement (2) of Lemma 3.1 holds for As; 474 Ars Math. Contemp. 17(2019)467-479 (2) If rk(AS) = 2 then Aut(A)S = Sym(S). Proof. Statement (1) of the lemma follows from [8, Corollary 4.4] and Statement (2) of the lemma follows from [8, Theorem 4.6 (1)]. □ Lemma 3.3. Suppose that Statement (2) of Lemma 3.1 holds for A. Then Aut(A) is 2-isolated. Proof. By [15, Lemma 8.2], it suffices to prove that Aut(A)e has a faithful regular orbit. The S-ring A is normal. So (2.1) implies that Aut(A)e < Aut(G). Let X G S(A) be highest. Since A is cyclotomic, each element of X is a generator of G. If f G Aut(A)e fixes some x G X then f is trivial because f G Aut(G) and x is a generator of G. Besides, A is schurian and hence X G Orb(Aut(A)e, G) by (2.2). Therefore X is a regular orbit of Aut(A)e. The group Aut(G) is cyclic because p is odd. So both of the groups Aut(A)e and Aut(A)X are cyclic groups of order |X |. Thus, X is a faithful regular orbit of Aut(A)e and the lemma is proved. □ Lemma 3.4. Suppose that Statement (2) of Lemma 3.1 holds for A and ^ is an algebraic isomorphism from A to an S-ring A' over an abelian group G'. Then G' is cyclic. Proof. By the hypothesis, A = Cyc(K, G) for some K < Aut(G) with |K| < p - 1. The group E = {g G G : |g| = p} is an A-subgroup of order p because A is cyclotomic. The group E' = Ev is an A'-subgroup of order p by the properties of an algebraic isomorphism. Assume that G' is non-cyclic. Then there exists X' G S(A') containing an element of order p outside E'. Let X g S(A) such that Xv = X'. The set X consists of elements of order greater than p because G is cyclic and all elements of order p from G lie inside E. The identity element e of G enters the element with a coefficient dividing by p because xp = e for each x G X. The identity element e' of G' enters the element (X')p with a coefficient which is not divided by p because (x')p = e' for some x' G X' and |X'| < p - 1. Since ^ is an algebraic isomorphism, we have (XT = (X')p and {e}v = {e'}. This implies that e and e' must enter and (X')p respectively with the same coefficients, a contradiction. Therefore G' is cyclic and the lemma is proved. □ Lemma 3.5. Suppose that | rad(A)| > 1. Then there exists an A-section S = U/L such that A is the proper S-wreath product, | rad(Ay) | = 1, and |L| = p. Proof. From [17, Lemma 5.2] it follows that there exists an A-section U/Li such that A is the proper U/L1-wreath product and | rad(Ay )| = 1. Let L be a subgroup of L1 of order p. Then the lemma holds for S = U/L. □ Lemma 3.6 ([5, Theorem 1.3]). Every S-ring over a cyclic p-group is separable with respect to . G. Ryabov: On separable abelian p-groups 475 Proof of the Theorem 1.1. The statement of the theorem for p e {2, 3} was proved in [17, Lemma 5.5]. Further we assume thatp > 5. Let A be an S-ring over a cyclicp-group G of order pk, where k > 1. Prove that A is separable. We proceed by induction on k. If k = 1 then G is the unique up to isomorphism group of order p and the statement of the theorem follows from Lemma 3.6. Let k > 2. One of the statements of Lemma 3.1 holds for A. If Statement (1) of Lemma 3.1 holds for A then rk(A) = 2 and hence A is separable. Suppose that Statement (2) of Lemma 3.1 holds for A. Let p be an algebraic isomorphism from A to an S-ring A' over an abelian group G'. Due to Lemma 3.4, the group G' is cyclic. So p is induced by an isomorphism by Lemma 3.6. Therefore A is separable. Now suppose that Statement (3) of Lemma 3.1 holds for A. Then A = Au lS AG/L for some A-section S = U/L with L > e and U < G. The S-rings Au and AG/L are separable by the induction hypothesis. Due to Lemma 3.5 we may assume that rad(AU) = e and |L| = p. In this case rk(AU) =2 or Statement (2) of Lemma 3.1 holds for Au. If rk(AU) =2 or |S| = 1 then U = L and A is separable by Lemma 2.3. Assume that Statement (2) of Lemma 3.1 holds for Au. If |S| > p2 then Statement (2) of Lemma 3.1 holds for AS by Statement (1) of Lemma 3.2. Lemma 3.3 implies that Aut(AS) is 2-isolated. The S-ring Au is cyclotomic and hence it is schurian. Therefore A is separable by Lemma 2.4. It remains to consider the case when |S| = p. In this case |U| = p2. If rad(X) > L for every X e S(A) outside U then rad(X) > U for every X e S(A) outside U because G is cyclic. This yields that A = Au l AG/U and hence A is separable by Lemma 2.3. Suppose that there exists X e S(A) outside U with rad(X) = L. The remaining part of the proof is divided into two cases. Case 1: (X} < G. In this case put Si = (X}/L. The S-ring A is the Si-wreath product and |S1| > p2. Note that | rad(ASl) | = 1 because rad(X) = L. So Statement (1) or Statement (2) of Lemma 3.1 holds for ASl. In the former case Aut(A^X) )Sl = Aut(ASl) = Sym(S1) by Statement (2) of Lemma 3.2 and A is separable by Lemma 2.3. In the latter case Aut(ASl) is 2-isolated by Lemma 3.3. Since A^ is schurian, the conditions of Lemma 2.4 hold for S1 and A is separable by Lemma 2.4. Case 2: (X} = G. In this case | rad(AG/L) | = 1 because rad(X) = L. Let n: G ^ G/L be the canonical epimorphism. Clearly, n(U) is an AG/L-subgroup and n(X) lies outside n(U). So rk(AG/L) > 2 and hence Statement (2) of Lemma 3.1 holds for AG/L. Let p be an algebraic isomorphism from A to an S-ring A' over an abelian group G'. Put U' = U^ and L' = Lv. Clearly, L' < U'. (3.1) The algebraic isomorphism p induces the algebraic isomorphism pu from Au to Au>, where U' = UFrom Lemma 3.4 it follows that U' = Cp2. (3.2) Also p induces the algebraic isomorphism pG/L from AG/L to AG//L. Lemma 3.4 implies that G'/L' is cyclic. Since |L'| = |L| = p, we conclude that G = Cpk or G = Cp x Cpk-l. 476 Ars Math. Contemp. 17(2019)467-479 However, in the latter case L' is not contained in a cyclic group of order p2 because G'/L' is cyclic. This contradicts to (3.1) and (3.2). So G' is cyclic and p is induced by an isomorphism by Lemma 3.6. Therefore A is separable and the theorem is proved. □ 4 Proof of Theorem 1.2 Proposition 4.1. The group Cp is separable for p e {2, 3}. Before we prove Propostion 4.1 we give the lemma providing a description of S-rings over these groups. Lemma 4.2. Let A be an S-ring over C:^, where p e {2,3}. Then A is schurian and one of the following statements holds: (1) rk(A) = 2; (2) A is the tensor product of smaller S-rings; (3) A is the proper S-wreath product of two S-rings with |S| < p; (4) p = 3 and A =Cay Aj, where Ai is one of the 14 exceptional S-rings whose parameters are listed in Table 1. Remark 4.3. In Table 1 the notation km means that an S-ring have exactly m basic sets of size k. Table 1: Parameters of the 14 exceptional S-rings Ai, A2,..., A14. S-ring rank sizes of basic sets Ai 3 1, 132 A2 4 1, 6, 8, 12 A3 4 1, 2, 122 a4 5 1, 42, 6, 12 A5 5 1, 2, 83 Ae 6 1, 2, 64 a7 7 1, 2, 44, 8 Ag 7 1, 2, 32, 63 A9 8 1, 2, 4e A10 9 1, 23, 45 Aii 10 1, 25, 44 A12 10 13, 3e, 6 A13 11 13, 38 A14 14 1, 213 Proof. The statement of the lemma can be checked with the help of the GAP package COCO2P [11]. □ G. Ryabov: On separable abelian p-groups 477 Proof of the Proposition 4.1. From [17, Theorem 1, Lemma 5.5] it follows that the group Cp is separable for p e {2,3} and k < 2. Let A be an S-ring over G = Cp, where p e {2,3}. Then one of the statements of Lemma 4.2 holds for A. If Statement (1) of Lemma 4.2 holds for A then, obviously, A is separable. If Statement (2) of Lemma 4.2 holds for A then A is separable by Lemma 2.2. Suppose that Statement (3) of Lemma 4.2 holds for A. Then A is the proper schurian S-wreath product for some A-section S = U/L with |S| < 3. Since A is schurian, Av is also schurian. Note that Aut(AS) is 2-isolated becasue |S| < 3. Therefore A is separable by Lemma 2.4. Suppose that Statement (4) of Lemma 4.2 holds for A and y is an algebraic isomorphism from A to an S-ring A' over an abelian group G'. Clearly, if A' is separable then y-1 is induced by an isomorphism and hence y is also induced by an isomorphism. If G' = Cp 3 then A' is separable by Theorem 1.1; if G' = Cp x Cp2 then A' is separable by [17, Theorem 1]; if G' = Cp and one of the Statements (1)-(3) of Lemma 4.2 holds for A' then A' is separable by the previous paragraph. So in the above cases y is induced by an isomorphism. Thus, we may assume that G' = Cp and Statement (4) of Lemma 4.2 holds for A'. Two algebraically isomorphic S-rings have the same rank and sizes of basic sets. So information from Table 1 implies that A¿ ^Alg Aj whenever i = j. Therefore we may assume that == == 2. Proof. In view of Lemma 2.5 to prove that the group Cp x Cp x Cpk is non-separable for p e {2,3} and k > 2 it is sufficient to construct an S-ring A over Cp x Cp x Cp2, p e {2, 3}, and an algebraic isomorphism y from A to itself which is not induced by an isomorphism. Let G = (a) x (6) x (c), where |a| = |6| = p and |c| = p2. Put A = (a), B = (6), C = (c), c1 = cp, and C1 = (c1). Firstly consider the case p = 2. Let f e Aut(G) such that f: (a, 6, c) ^ (a, 6ac1, ca) and A = Cyc((f), G). It easy to see that |f | =2 and the basic sets of A are the following To = {e}, T1 = {a}, T2 = {c1}, T3 = {acj, X1 = cA, X2 = c3A, Y1 = 6(ac1), Y2 = 6a(ac1), Z1 = 6cC1, Z2 = 6caC1. Define a permutation y on the set S(A) as follows: TT = T0, Tf = T1, Tf = T3, Tf = T2, X? = X1, Xff = X2, 478 Ars Math. Contemp. 17(2019)467-479 Y1 = Zi, Y2f = Z2, Zf = Yi, Z2f = Y2. It easy to see that |p| = 2. The straightforward check implies that p is an algebraic t tv isomorphism from A to itself. Let us check, for example, that cy2 = cy2^ yv. We have Y1Y2 = 2a + 2c1 and YfYf = Z1Z2 = 2a + 2ac1. So cT, Y2 = c^ Y ^ = 2. Note that A corresponds to a Kleinian quasi-thin scheme of index 4 in the sense of [14]. The S-ring A is cyclotomic and hence it is schurian. Assume that p is induced by an isomorphism. Then the algebraic fusion A^ is schurian by Lemma 2.1. However, computer calculations made by using the package COCO2P [11] (see also [21]) imply that Al