¿^creative ^commor ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 19 (2020) 277-295 https://doi.org/10.26493/1855-3974.2348.f42 (Also available at http://amc-journal.eu) ARS MATHEMATICA CONTEMPORANEA The Cayley isomorphism property '5 x Cp for the group C5 x Cp Grigory Ryabov * © Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia, and Novosibirsk State University, 1 Pirogova st., 630090, Novosibirsk, Russia Received 28 May 2020, accepted 29 July 2020, published online 17 November 2020 Abstract A finite group G is called a DCI-group if two Cayley digraphs over G are isomorphic if and only if their connection sets are conjugate by a group automorphism. We prove that the group Cf x Cp, where p is a prime, is a DCI-group if and only if p = 2. Together with the previously obtained results, this implies that a group G of order 32p, where p is a prime, is a DCI-group if and only if p = 2 and G = Cf x Cp. Keywords: Isomorphisms, DCl-groups, Schur rings. Math. Subj. Class. (2020): 05C25, 05C60, 20B25 1 Introduction Let G be a finite group and S C G. The Cayley digraph Cay(G, S) over G with connection set S is defined to be the digraph with vertex set G and arc set {(g, sg) : g G G, s g S}. Two Cayley digraphs over G are called Cayley isomorphic if there exists an isomorphism between them which is also an automorphism of G. Clearly, two Cayley isomorphic Cayley digraphs are isomorphic. The converse statement is not true in general (see [3, 10]). A subset S C G is called a CCI-subset if for each T C G the Cayley digraphs Cay(G, S) and Cay(G, T) are isomorphic if and only if they are Cayley isomorphic. A finite group G is called a DCI-group (CI-group, respectively) if each subset of G (each inverse-closed subset of G, respectively) is a CI-subset. *The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613 with the Ministry of Science and Higher Education of the Russian Federation. The author would like to thank Prof. Istvan Kovacs for the fruitful discussions on the subject matters, Prof. Pablo Spiga and the anonymous referee for valuable comments which help to improve the text significantly. E-mail address: gric2ryabov@gmail.com (Grigory Ryabov) ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 277 Ars Math. Contemp. 19 (2020) 189-208 The investigation of DCI-groups was initiated by Adam [1] who conjectured, in our terms, that every cyclic group is a DCI-group. This conjecture was disproved by Elspas and Turner in [10]. The problem of determining of finite DCI- and Cl-groups was suggested by Babai and Frankl in [5]. For more information on DCI- and Cl-groups we refer the readers to the survey paper [21]. In this paper we are interested in abelian DCI-groups. The cyclic group of order n is denoted by Cn. Elspas and Turner [10] and independently Djokovic [8] proved that every cyclic group of prime order is a DCI-group. The fact that Cpq is a DCI-group for distinct primes p and q was proved by Alspach and Parsons in [3] and independently by Klin and Poschel in [17]. The complete classification of cyclic DCI-groups was obtained by Muzychuk in [23, 24]. He proved that a cyclic group of order n is a DCI-group if and only if n = k or n = 2k, where k is square-free. Denote the class of all finite abelian groups where every Sylow subgroup is elementary abelian by E. From [18, Theorem 1.1] it follows that every DCI-group is the coprime product (i.e. the direct product of groups of coprime orders) of groups from the following list: Cpk, C4, Q8, A4, H x (z), where p isaprime, H is a group of odd order from E, |z| G {2,4}, and hz = h-1 for every h G H. One can check that the class of DCI-groups is closed under taking subgroups. So one of the crucial steps towards the classification of all DCI-groups is to determine which groups from E are DCI. The following non-cyclic groups from E are DCI-groups (p and q are assumed to be distinct primes): Cp [2, 14]; Cp [ , 9]; C4, Cf [ ]; Cp, where p is odd [15] (a proof for Cp with no condition on p was given in [ ]); Cp, where p is odd [ 13]; Cp x Cq [18 ]; Cp x Cq [ ]; Cp4 x Cq [ ]. The smallest example of a non-DCI-group from E was found by Nowitz [28]. He proved that C6 is non-DCI. This implies that C^ is non-DCI for every n > 6. Also C3n is non-DCI for every n > 8 [33] and C^ is non-DCI for every prime p and n > 2p + 3[ ]. In this paper we find a new infinite family of DCI-groups from E which are close to the smallest non-DCI-group from E. The main result of the paper can be formulated as follows. Theorem 1.1. Let p be a prime. Then the group Cf x Cp is a DCI-group if and only if p = 2. Theorem 1.1 extends the results obtained in [18, 20, 27] which imply that the group Cp x Cq is a DCI-group whenever p and q are distinct primes and k < 4. Note that the "only if" part of Theorem 1.1, in fact, was proved by Nowitz in [28]. The next corollary immediately follows from [18, Theorem 1.1] and Theorem 1.1. Corollary 1.2. Let p be a prime. Then a group G of order 32p is a DCI-group if and only ifp = 2 and G = Cf x Cp. To prove Theorem 1.1, we use the S-ring approach. An S-ring over a group G is a subring of the group ring ZG which is a free Z-module spanned by a special partition of G. If every S-ring from a certain family of S-rings over G is a CI-S-ring then G is a DCI-group (see Section 4). The definition of an S-ring goes back to Schur [31] and Wielandt [34]. The usage of S-rings in the investigation of DCI-groups was proposed by G. Ryabov: The Cayley isomorphism property for the group C| x Cp 279 Klin and Poschel [17]. Most recent results on DCI-groups were obtained using S-rings (see [15, 18, 19, 20, 27]). The text of the paper is organized in the following way. In Section 2 we provide definitions and basic facts concerned with S-rings. Section 3 contains a necessary information on isomorphisms of S-rings. In Section 4 we discuss CI-S-rings and their relation with DCI-groups. We also prove in this section a sufficient condition of CI-property for S-rings (Lemma 4.4). Section 5 is devoted to the generalized wreath and star products of S -rings. Here we deduce from previously obtained results two sufficient conditions for the generalized wreath product of S-rings to be a CI-S-ring (Lemma 5.5 and Lemma 5.8). Section 6 and 7 are concerned with p-S-rings and S-rings over a group of order pk, where p is a prime and GCD(p, k) = 1, (so-called non-powerful order) respectively. In Section 8 we provide properties of S-rings over the groups C^, n < 5, and prove that all S-rings over these groups are CI. The material of this section is based on computational results obtained with the help of the GAP package COCO2P [16]. Finally, in Section 9 we prove Theorem 1.1. Notation. Let G be a finite group and X Ç G. The element J2xeX x of the group ring ZG is denoted by X. 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}. Given a set X Ç G the set {(g, xg) : x G X, g G G} of arcs of the Cayley digraph Cay(G, X) is denoted by A(X). The group of all permutations of G is denoted by Sym(G). The subgroup of Sym(G) consisting of all right translations of G is denoted by Gright. The set {K < Sym(G) : K > Gright} is denoted by Sup(Gright). For a set A Ç 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 K < Sym(Q) and a G Q then the stabilizer of a in K and the set of all orbits of K on Q are denoted by Ka and Orb(K, Q) respectively. If H < G then the normalizer of H in G is denoted by NG(H). The cyclic group of order n is denoted by Cn. The class of all finite abelian groups where every Sylow subgroup is elementary abelian is denoted by E. 2 S-rings In this section we give a background of S-rings. In general, we follow [20], where the most part of the material is contained. For more information on S-rings we refer the readers to [6, 25]. Let G be a finite group and ZG the integer group ring. Denote the identity element of G by e. A subring A C ZG is called an S-ring (a Schur ring) over G if there exists a partition S(A) of G such that: (1) {e} G S(A), (2) if X G S(A) then X-1 G S(A), (3) A = SpanZ{X : X G S (A)}. 280 Ars Math. Contemp. 19 (2020) 189-208 The elements of S(A) are called the basic sets of A and the number rk(A) = |S(A)| is called the rank of A. If X, Y e S(A) then XY e S(A) whenever |X | = 1 or |Y | = 1. Let A be an S-ring over a group G. A set X C G is called an A-set if X e A. A subgroup H < G is called an A-subgroup if H is an A-set. From the definition it follows that the intersection of A-subgroups is also an A-subgroup. One can check that for each A-set X the groups (X} and rad(X) are A-subgroups. By the thin radical of A we mean the set defined as Oe(A) = {x G G : {x} e S(A)}. It is easy to see that Oe (A) is an A-subgroup. Lemma 2.1 ([11, Lemma 2.1]). Let A be an S-ring over a group G, H an A-subgroup of G, and X e S(A). Then the number |X n Hx| does not depend on x e X. Let L < U < G. A section U/L is called 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 e S(A), X C U} , where n: U ^ U/L is the canonical epimorphism, is an S-ring over S. 3 Isomorphisms and schurity Let A and A' be S-rings over groups G and G' respectively. A bijection f: G ^ G' is called an isomorphism from A to A' if {A(X)f : X e S(A)} = {A(X') : X' e S(A')}, where A(X)f = {(gf, hf) : (g, h) e A(X)}. If there exists an isomorphism from A to A' then we say that A and A' are isomorphic and write A = A'. The group of all isomorphisms from A onto itself contains a normal subgroup {f e Sym(G) : A(X)f = A(X) for every X e S(A)} called the automorphism group of A and denoted by Aut(A). The definition implies that Gright < Aut(A). The S-ring A is called normal if Gright is normal in Aut(A). One can verify that if S is an A-section then Aut(A)S < Aut(AS). Denote the group Aut(A) n Aut(G) by AutG(A). It easy to check that if S is an A-section then AutG(A)S < AutS (AS). One can verify that AutG(A) = (NAut(A)(Gright))e. Let K e Sup(Gright). Schur proved in [31] that the Z-submodule V(K, G) = SpanZ{X : X e 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 e Sup(Gright). One can verify that given Ki, K2 e Sup(Gright), if Ki < K2 then V(Ki,G) > V(K2,G). (3.1) G. Ryabov: The Cayley isomorphism property for the group C| x Cp 281 If A = V (K, G) for some K e Sup(Gright) and S is an A-section then AS = V (KS, S). So if A is schurian then AS is also schurian for every A-section S. It can be checked that V(Aut(A), G) > A (3.2) and the equality is attained if and only if A is schurian. An S-ring A over a group G is defined to be cyclotomic if there exists K < Aut(G) such that S(A) = Orb(K, G). In this case we write A = Cyc(K, G). Obviously, A = V(GrightK, G). So every cyclotomic S-ring is schurian. If A = Cyc(K, G) for some K < Aut(G) and S is an A-section then AS = Cyc(KS, S). Therefore if A is cyclotomic then AS is also cyclotomic for every A-section S. Two permutation groups Ki and K2 on a set Q are called 2-equivalent if Orb(K, Q2) = Orb(K2, Q2) (here we assume that K1 and K2 act on Q2 componentwise). In this case we write K1 «2 K2. The relation «2 is an equivalence relation on the set of all subgroups of Sym(Q). Every equivalence class has a unique maximal element with respect to inclusion. Given K < Sym(Q), this element is called the 2-closure of K and denoted by K(2). If A = V(K, G) for some K e Sup(Gright) then K(2) = Aut(A). An S-ring A over G is called 2-minimal if {K e Sup(Gright) : K «2 Aut(A)} = {Aut(A)}. Two groups K1,K2 < Aut(G) are said to be Cayley equivalent if Orb(K1,G) = Orb(K2, G). In this case we write K1 «Cay K2. If A = Cyc(K, G) for some K < Aut(G) then AutG (A) is the largest group which is Cayley equivalent to K. A cyclotomic S-ring A over G is called Cayley minimal if {K < Aut(G) : K «cay AutG(A)} = {AutG(A)}. It is easy to see that ZG is 2-minimal and Cayley minimal. 4 CI-S-rings Let A be an S-ring over a group G. Put Iso(A) = {f e Sym(G) : f is an isomorphism from A onto an S-ring over G}. One can see that Aut(A) Aut(G) C Iso(A). However, the converse statement does not hold in general. The S-ring A is defined to be a CI-S-ring if Aut(A) Aut(G) = Iso(A). It is easy to check that ZG and the S-ring of rank 2 over G are CI-S-rings. Put Sup2 (Gright) = {K e Sup(Gright) : K(2) = K}. The group M < Sym(G) is said to be G-regular if M is regular and isomorphic to G. Following [15], we say that a group K e Sup(Gright) is G-transjugate if every G-regular subgroup of K is K-conjugate to Gright. Babai proved in [4] the statement which can be formulated in our terms as follows: a set S C G is a CI-subset if and only if the group Aut(Cay(G, S)) is G-transjugate. The next lemma provides a similar criterion for a schurian S-ring to be CI. Lemma 4.1. Let K e Sup2(Gright) and A = V (K, G). Then A is a CI-S-ring if and only if K is G-transjugate. 282 Ars Math. Contemp. 19 (2020) 189-208 Proof. The statement of the lemma follows from [15, Theorem 2.6]. □ Let Ki,K2 G Sup(Gright) such that K < K2. Then K is called a G-complete subgroup of K2 if every G-regular subgroup of K2 is K2-conjugate to some G-regular subgroup of Ki (see [15, Definition 2]). In this case we write Ki V(Aut(A), G) > A. So U and L are also B-subgroups. Let C = ZU \s Z(G/L). The S-rings Gu and CG/L are schurian and Cs is 2-minimal because Cs = ZS. So G is schurian by [26, Corollary 10.3]. This implies that G = V(Aut(G), G). (5.1) Every element from Aut(C)e fixes every basic set of G and hence it fixes every L-coset. Since L1 > L, every element from Aut(C)e fixes every Li-coset. We conclude that 284 Ars Math. Contemp. 19 (2020) 189-208 Aut(C)e < Aut(A)G/Ll and hence Aut(C) < K. Now from Equations (3.1) and (5.1) it follows that e = V(Aut(e), G) > V(K, G) = B. (5.2) The group U is a B- and a C-subgroup. Due to Equation (5.2), every basic set of B which lies outside U is a union of some basic sets of C which lie outside U .So L < rad(X) for every X G S(B) outside U. Thus, B is the S-wreath product. □ Lemma 5.5. In the conditions of Lemma 5.1, suppose that: (1) every S-ring over U is a CI-S-ring; (2) AG/L is 2-minimal or normal. Then A is a CI-S-ring. Proof. Let B = V(K, G), where K = Aut(A)G/LGright. From Lemma 5.4 it follows that B is the S-wreath product. Since L1 = L, the definition of B implies that BG/L = Z(G/L) and hence BS = ZS. Clearly, BG/L is a CI-S-ring. The S-ring Bv is a CI-S-ring by the assumption of the lemma. Therefore B is a CI-S-ring by Lemma 5.2. The S-ring AG/L is a CI-S-ring by the assumption of the lemma. Thus, A is a CI-S-ring by [20, Lemma 3.6] whenever AG/L is 2-minimal and by Lemma 4.4 whenever AG/L is normal. □ Let V and W be A-subgroups. The S-ring A is called the star product of AV and AW if the following conditions hold: (1) V n W < W; (2) each T g S(A) with T C (W \ V) is a union of some V n W-cosets; (3) for each T g S(A) with T C G \ (V U W) there exist R G S(Av) and S G S(Aw) such that T = RS. In this case we write A = AV * AW. The construction of the star product of S-rings was introduced in [15]. The star product is called nontrivial if V = {e} and V = G. If V n W = {e} then the star product is the usual tensor product of AV and AW (see [ 1, p. 5]). In this case we write A = AV ( AW. One can check that if A = AV ( AW then Aut(A) = Aut(AV) x Aut(AW). If V n W = {e} then A is the nontrivial V/(V n W)-wreath product. Indeed, let T g S(A) such that T £ V .If T C W \ V then V n W < rad(T) by Condition (2) of the definition. If T C G \ (V U W) then T = RS for some R G S(AV) and some S g S(Aw ) such that S C W \ V by Condition (3) of the definition. Since V n W < rad(S), we obtain V n W < rad(T). Lemma 5.6. Let G G E and A a schurian S-ring over G. Suppose that A = AV * AW for some A-subgroups V and W of G and the S-rings AV and AW/(VnW) are CI-S-rings. Then A is a CI-S-ring. Proof. The statement of the lemma follows from [18, Proposition 3.2, Theorem 4.1]. □ Lemma 5.7 ([13, Lemma 2.8]). Let A be an S-ring over an abelian group G — Gi x G2. Assume that G1 and G2 are A-groups. Then A = AGl ( AG2 whenever AGl or AG2 is the group ring. Lemma 5.8. In the conditions of Lemma 5.1, suppose that |G : U | is a prime and there exists X G S(Ag/l) outside S with |X| = 1. Then A is a CI-S-ring. G. Ryabov: The Cayley isomorphism property for the group C| x Cp 285 Proof. Let X = {x} for some x G G/L. Due to G G E, we conclude that |(x)| is prime. So |(x) n S| = 1 because x lies outside S. Since |G : U| is a prime, G/L = (x) x S. Note that A^ = Z(x). Therefore AG/L = Z(x) As by Lemma 5.7. Let ^ G Auts(As). Define ^ G Aut(G/L) in the following way: = x^ = x. Then ^ g AutG/L(AG/L) because AG/L = Z(x) AS. We obtain that AutG/i(AG/i)S > Auts(As), and hence AutG/L(AG/L)S = Auts(As). Thus, A is a Cl-S-ring by Lemma 5.1. □ 6 p-S-rings Let p be a prime. An S-ring A over a p-group G is called a p-S-ring if every basic set of A has a p-power size. Clearly, if |G| = p then A = ZG. In the next three lemmas G is a p-group and A is a p-S-ring over G. Lemma 6.1. If B > A then B is a p-S-ring. Proof. The statement of the lemma follows from [29, Theorem 1.1]. □ Lemma 6.2. Let S = U/L be an A-section of G. Then As is a p-S-ring. Proof. From Lemma 2.1 it follows that for every X G S(A) the number A = |X n Lx| does not depend on x g X .So A divides |X | and hence A is a p-power. Let n: G ^ G/L be the canonical epimorphism. Note that |n(X )| = |X |/A and hence |n(X )| is a p-power. Therefore every basic set of As has a p-power size. Thus, As is a p-S-ring. □ Lemma 6.3 ([13, Proposition 2.13]). The following statements hold: (1) |Oe(A)| > 1; (2) there exists a chain of A-subgroups {e} = G0 < Gi < • • • < Gs = G such that |Gj+i : Gj| = p for every i G {0,..., s - 1}. Lemma 6.4. Let G be an abelian group, K G Supmin(Grlght), and A = V(K, G). Suppose that H is an A-subgroup of G such that G/H is a p-group for some prime p. Then Ag/h is a p-S-ring. Proof. The statement of the lemma follows from [18, Lemma 5.2]. □ 7 S-rings over an abelian group of non-powerful order A number n is called powerful if p2 divides n for every prime divisor p of n. From now throughout this section G = H x P, where H is an abelian group and P = Cp, where p is a prime coprime to | H|. Clearly, |G| is non-powerful. Let A be an S-ring over G, Hi a maximal A-subgroup contained in H, and Pi the least A-subgroup containing P. Note that Hi Pi is an A-subgroup. 286 Ars Math. Contemp. 19 (2020) 189-208 Lemma 7.1 ([20, Lemma 6.3]). In the above notations, if H = (Hi-Pi)p/, the Hall p'-subgroup of H1P1, then AHlPl = AHl * APl. Lemma 7.2 ([27, Proposition 15]). In the above notations, if AHlPl/Hl = ZCp then AHi Pi = Ahi * APi. Lemma 7.3 ([11, Lemma 6.2]). In the above notations, suppose that H1 < H. Then one of the following statements holds: (1) A = Ahi I AG/Hi with rk(AG/Hi) = 2; (2) A = Ah1p1 Is AG/Pl, where S = Hi Pi/Pi and Pi < G. 8 S-rings over CJ,1, n < 5 All S-rings over the groups C2\ where n < 5, were enumerated with the help of the GAP package COCO2P [16]. The list of all S-rings over these groups is available on the web-page [30] (see also [35]). The next lemma is an immediate consequence of the above computational results (see also [11, Theorem 1.2]). Lemma 8.1. Every S-ring over Clf, where n < 5, is schurian. To prove Theorem 1.1, we will show that every schurian S-ring over Cf x Cp is CI. Since the most of schurian S-rings over Cf x Cp are generalized wreath or star products of S-rings over its proper subgroups, we need to check that all schurian S-rings over proper subgroups of Cf x Cp are CI. In this section we will do it for G = Cn, where n < 5. Note that G is a DCI-group by [2, 7] but this does not imply that every S-ring over G is CI (see Remark 4.3). We will describe 2-S-rings over G using computational results and check that all S-rings over G are CI. Until the end of the section G is an elementary abelian 2-group of rank n and A is a 2-S-ring over G. Lemma 8.2. Let n < 3. Then A is cyclotomic. Moreover, A is Cayley minimal except for the case when n = 3 and A = ZC21 ZC21ZC2. Proof. The first part of the lemma follows from [20, Lemma 5.2]; the second part follows from [20, Lemma 5.3]. □ Analyzing the lists of all S-rings over C2 and Cf available on the web-page [30], we conclude that up to isomorphism there are exactly nineteen 2-S-rings over G if n = 4 and there are exactly one hundred 2-S-rings over G if n = 5. It can can be established by inspecting the above 2-S-rings one after the other that there are exactly fifteen decomposable and four indecomposable 2-S-rings over G if n = 4 and there are exactly ninety six decomposable and four indecomposable 2-S-rings over G if n = 5. Lemma 8.3. Let n G {4,5} and A indecomposable. Then A is normal. If in addition n = 5 then A = ZC2 (g> .A', where A' is indecomposable 2-S-ring over C2. Proof. Let n = 4. One can compute | Aut(A)| and |NAut(A)(Gright)| using the GAP package COCO2P [16]. It turns out that for each of the four indecomposable 2-S-rings over G the equality | Aut(A)| = |NAut(A)(Gright)| G. Ryabov: The Cayley isomorphism property for the group C| x Cp 287 is attained. So every indecomposable 2-S-ring over G is normal whenever n = 4. Let n = 5. The straightforward check for each of the four indecomposable 2-S-rings over G yields that A = AH ( ZL, where H = C|, L = C2, and AH is indecomposable 2-S-ring. Clearly, ZL is normal. By the above paragraph, AH is normal. Since Aut(A) = Aut(AH) x Aut(AL), we obtain that A is normal. □ Note that if p > 2 then Lemma 8.3 does not hold. In fact, if p > 2 then there exists an indecomposable p-S-ring over Cp which is not normal (see [ , Lemma 6.4]). Lemma 8.4. Let n < 5. Then A is normal whenever one of the following statements holds: (1) A is indecomposable; (2) |G : Og (A)| = 2; (3) n = 4 and A = (ZC21 ZC2) © (ZC2 \ ZC2). Proof. If n < 3 and A is indecomposable then A = ZG by [ , Lemma 5.2]. Clearly, in this case A is normal. If n G {4, 5} and A is indecomposable then A is normal by Lemma 8.3. There are exactly n — 1 2-S-rings over G for which Statement (2) of the lemma holds. For every A isomorphic to one of these 2-S-rings and for A = (ZC21ZC2) ( (ZC2 I ZC2) one can compute | Aut(A)| and |NAut(A)(Gright)| using the GAP package COCO2P [16]. It turns out that in each case the equality | Aut(A)| = |NAut(A)(Gright)| holds and hence A is normal. □ Lemma 8.5. Let n = 4. Then A is cyclotomic. Proof. If A is decomposable then A is cyclotomic by [20, Lemma 5.6]. If A is indecomposable then A is normal by Lemma 8.3. This implies that Aut(A)e = (NAut(A)(Gright))e < Aut(G). The S-ring A is schurian by Lemma 8.1. So from Equation (3.2) it follows that A = V(Aut(A), G) and hence A = Cyc(Aut(A)e, G). □ Lemma 8.6. Let n = 5. Suppose that A is decomposable and |Og (A)| = 8. Then A is cyclotomic. Proof. Let A be the nontrivial S-wreath product for some A-section S = U/L. Note that |U| < 16, |G/L| < 16, and |S| < 8. The S-rings Au, AG/L, and As are 2-S-rings by Lemma 6.2. So each of these S-rings is cyclotomic by Lemma 8.2 whenever the order of the corresponding group is at most 8 and by Lemma 8.5 otherwise. Since | Og (A) | = 8, we conclude that |S| < 4 or |S| = 8 and |Og(AS) | > 4. In both cases AS is Cayley minimal by Lemma 8.2. This implies that Autu (Au )S = AutG/i(AG/i)S = Auts (As ). Now from [20, Lemma 4.3] it follows that A is cyclotomic. □ 288 Ars Math. Contemp. 19 (2020) 189-208 In the next two lemmas we establish some properties of decomposable 2-S-rings over G = Cf whose thin radical is of size 2 or 4. These properties will be used in the proof of Theorem 1.1. The statements of Lemma 8.7 and Lemma 8.8 can be verified by analysis of computational results obtained with the help of the GAP package COCO2P [16]. For every decomposable 2-S-ring A with |Oo(A) | g {2,4} over G (see the list [30]), we compute all A-subgroups, automorphism groups, and Cayley automorphism groups of some restrictions and quotients. Lemma 8.7. Let n = 5. Suppose that A is decomposable and | Oo (A) | = 4. Then one of the following statements holds: (1) there exists an A-subgroup L < Oo (A) of order 2 such that A = ZOo (A) ls AG/L, where S = Oe (A)/L; (2) | AutG(A)| > | Auty(Au)| for every A-subgroup U with |U| = 16 and U > Oo (A); (3) A is normal; (4) there exist an A-subgroup L < Oo(A) and X g S(A) such that |L| = |X| = 2, L = rad(X), and AG/L is normal. Lemma 8.8. Let n = 5. Suppose that A is decomposable, |Oo(A)| = 2, and there exists X g S(A) with |X| > 1 and | rad(X)| = 1. Then |X| = 4 and one of the following statements holds: (1) A = B l ZC2, where B is a 2-S-ring over C|; (2) | AutG(A)| > | Autu(Au)| for every A-subgroup U with |U| = 16; (3) there exists an A-subgroup L such that |L| g {2,4} and AG/L is normal. Lemma 8.9. Let D g E such that every S-ring over a proper section of D is CI, D an S-ring over D, and S = U/L a D-section. Suppose that D is the nontrivial S-wreath product. Then D is a CI-S-ring whenever D/L = Ck for some k < 4 and Dd/l is a 2-S-ring. Proof. The S-ring Dd/l is cyclotomic by Lemma 8.2 whenever |D/L| < 8 and by Lemma 8.5 whenever |D/L| = 16. The S-ring Ds is a 2-S-ring by Lemma 6.2. If Ds £ ZC2 I ZC2 l ZC2 then Ds is Cayley minimal by Lemma 8.2. The S-rings Dy and Dd/l are CI-S-rings by the assumption of the lemma. So D is a CI-S-ring by Lemma 5.3. Assume that Ds = ZC21 ZC21 ZC2. In this case |D/L| = 16, |S| = 8, and there exists the least Ds-subgroup A of S of order 2. Every basic set of Dd/l outside S is contained in an S-coset because D(D/L)/s = ZC2. So rad(X) is a Ds-subgroup for every X g S(Dd/l) outside S. If | rad(X)| > 1 for every X g S (Dd/l ) outside S then Dd/l is the S/A-wreath product because A is the least Ds-subgroup. This implies that D is the U/n-1(A)-wreath product, where n: D ^ D/L is the canonical epimorphism. One can see that |D/n-1(A)| < 8 and |U/n-1(A)| < 4. The S-rings DD/n-i(A) and Dy/n-i(A) are 2-S-rings by Lemma 6.2. The S-ring DD/n-i(A) is cyclotomic by Lemma 8.2 and the S-ring Dy/n-i(A) is Cayley minimal by G. Ryabov: The Cayley isomorphism property for the group C| x Cp 289 Lemma 8.2. The S-rings Du and DD/n-i(A) are CI-S-rings by the assumption of the lemma. Thus, D is a Cl-S-ring by Lemma 5.3. Suppose that there exists a basic set X of Dd/l outside S with | rad(X)| = 1. If Dd/l is decomposable then AutD/L(DD/L)S = Auts (Ds ) by [20, Lemma 5.8]. Therefore D is a CI-S-ring by Lemma 5.1. If Dd/l is indecomposable then Dd/l is normal by Lemma 8.3. So all conditions of Lemma 5.5 hold for D. Thus, D is a CI-S-ring. □ Lemma 8.10. Let n < 5. Then every S-ring over G is a CI-S-ring. Proof. Every S-ring over G is schurian by Lemma 8.1. So to prove the lemma, it is sufficient to prove that B = V(K, G) is a CI-S-ring for every K e Supmm(Gright) (see Remark 4.3). The S-ring B is a 2-S-ring by Lemma 6.4. If n < 4 then B is CI by [20, Lemma 5.7]. Thus, if n = 4 then the statement of the lemma holds. Let n = 5. Suppose that B is indecomposable. Then the second part of Lemma 8.3 implies B = ZC2 B', where B' is indecomposable 2-S-ring over C2. Since B is schurian by Lemma 8.1 and every S-ring over an elementary abelian group of rank at most 4 is CI by the above paragraph, we conclude that B is a CI-S-ring by Lemma 5.6. Now suppose that B is decomposable, i.e. B is the nontrivial S = U/L-wreath product for some B-section S = U/L. Clearly, |G/L| < 16. The S-ring BG/L is a 2-S-ring by Lemma 6.2. Since every S-ring over an elementary abelian group of rank at most 4 is CI, B is a CI-S-ring by Lemma 8.9. □ 9 Proof of Theorem 1.1 Let G = H x P, where H = Cf and P = Cp, where p is a prime. These notations are valid until the end of the paper. If p = 2 then G is not a DCI-group by [ ]. So in view of Lemma 4.2, to prove Theorem 1.1, it is sufficient to prove the following theorem. Theorem 9.1. Let p be an odd prime and K e Supmm(Gright). Then .A = V(K, G) is a CI-S-ring. The proof of Proposition 9.1 will be given at the end of the section. We start with the next lemma concerned with proper sections of G. Lemma 9.2. Let S be a section of G such that S = G. Then every schurian S-ring over S is a CI-S-ring. Proof. If S = C2n for some n < 5 then we are done by Lemma 8.10. Suppose that S = C2n x Cp for some n < 4. Then the statement of the lemma follows from [20, Remark 3.4] whenever n < 3 and from [ , Remark 3.4, Theorem 7.1] whenever n = 4. □ A key step towards the proof of Theorem 9.1 is the following lemma. Lemma 9.3. Let A be an S-ring over G and U an A-subgroup with U > P. Suppose that P is an A-subgroup, A is the nontrivial S-wreath product, where S = U/P, |S| = 16, and Ag/p is a 2-S-ring. Then A is a CI-S-ring. 290 Ars Math. Contemp. 19 (2020) 189-208 Proof. Firstly we prove two lemmas concerned with some special cases of Lemma 9.3. Lemma 9.4. Suppose that S has a gwr-complement with respect to AG/P. Then A is a CI-S-ring. Proof. The condition of the lemma implies that there exists an AG/P-subgroup A such that AG/P is the nontrivial S/A-wreath product. This means that A is the nontrivial U/n-1(A)-wreath product, where n: G ^ G/P is the canonical epimorphism. Note that |G/n-1(A)| < 16 and AG/n-i(A) = A(G/P)/A is a 2-S-ring by Lemma 6.2. Therefore A is a CI-S-ring by Lemma 9.2 and Lemma 8.9. □ Lemma 9.5. Suppose that S does not have a gwr-complement with respect to AG/P. Then | AutG/p(AG/p)s| = | AutG/p(AG/p)|. Proof. To prove the lemma it is sufficient to prove that the group (AutG/p(Ag/p))s = W € AutG/p(Ag/p) : = ids} is trivial. Let ^ € (AutG/P(AG/P))s. Put C = Cyc((<^), G/P). Clearly, (<^) < Aut(AG/P). So from Equations (3.1) and (3.2) it follows that C > AG/P. Lemma 6.1 yields that C is a 2-S-ring. Since = idS, we conclude that Oe(C) > S. If C = Z(G/P) then Oe(C) = S. Therefore C = ZS ?SM Z((G/P)/A) for some C-subgroup A by Statement (i) of [ , Proposition 4.3]. This implies that AG/P = AS A((G/P)/A) because C > AG/P and S is both AG/P, C-subgroup. We obtain a contradiction with the assumption of the lemma. Thus, C = Z(G/P) and hence ^ is trivial. So the group (AutG/p(Ag/p))s is trivial. □ If Ag/p is indecomposable then AG/P is normal by Lemma 8.3. So A is a CI-S-ring by Lemma 9.2 and Lemma 5.5. Further we assume that AG/P is decomposable. Due to Lemma 9.4, we may assume also that S does not have a gwr-complement with respect to AG/P. (9.1) If there exists X € S(AG/P) outside S with |X | = 1 then A is a CI-S-ring by Lemma 9.2 and Lemma 5.8. So we may assume that Oe (Ag/p) < S. (9.2) Note that |Oe(AG/P)| > 1 by Statement (1) of Lemma 6.3 and |Oe(AG/P)| < 16 by Equation (9.2). So |Oe(AG/P)| € {2,4, 8,16}. We divide the rest of the proof into four cases depending on | Oe(AG/P) |. Case 1: |Oe(Ag/p)| = 16. Due to Equation (9.2), we conclude that AS = ZS. So A is a CI-S-ring by Lemma 9.2 and Lemma 5.2. Case 2: |Oe(Ag/p)| = 8. Since Ag/p is decomposable, Lemma 8.6 implies that AG/P is cyclotomic. The S-ring AS is a 2-S-ring by Lemma 6.2. In view of Equation (9.2), we obtain that |Oe (AS )| = 8. So Statement (ii) of [19, Proposition 4.3] yields that the S-ring AS is Cayley minimal. Thus, A is a CI-S-ring by Lemma 9.2 and Lemma 5.3. G. Ryabov: The Cayley isomorphism property for the group C| x Cp 291 Case3: |Oe(Ag/p)| = 4. In this case one of the statements of Lemma 8.7 holds for AG/P. If Statement (1) of Lemma 8.7 holds for AG/P then we obtain a contradiction with Equation (9.1). If Statement (2) of Lemma 8.7 holds for AG/P then | AutG/P (AG/P )| > | Auts (As )|. From Lemma 9.5 it follows that | AutG/P(AG/P)S| = | AutG/P(AG/P)| and hence | AutG/P(Ag/p)S| > | Auts(As)|. Since AutG/P(AG/P)s < Auts(As), we conclude that AutG/P(AG/P)s = Auts(As). Thus, A is a CI-S-ring by Lemma 9.2 and Lemma 5.1. If Statement (3) of Lemma 8.7 holds for AG/P then AG/P is normal. In this case A is a CI-S-ring by Lemma 9.2 and Lemma 5.5. Suppose that Statement (4) of Lemma 8.7 holds for AG/P, i.e. there exists an AG/P-subgroup A < Oe (AG/P) of order 2 and X = {x^x2} G S(AG/P) suchthat A(G/P )/A is normal and A = rad(X). Let L = n-1(A), where n: G ^ G/P is the canonical epimorphism, and B = V(N, G), where N = Aut(A)G/LGright. Prove that B is a CI-S-ring. Lemma 5.4 implies that B is the S-wreath product. From Equations (3.1) and (3.2) it follows that B > A. So BG/P > AG/P and hence BG/P is a 2-S-ring by Lemma 6.1. We obtain that B and U satisfy the conditions of Lemma 9.3. One can see that X is a BG/P-set and Oe(Bg/p) > Oe(Ag/p) (9.3) because BG/P > AG/P. The definition of B yields that every basic set of B is contained in an L-coset and hence every basic set of BG/P is contained in an A-coset. Therefore {xi}, {x2}G S(Bg/p) (9.4) because X is a BG/P-set and A = rad(X). Now from Equations (9.3) and (9.4) it follows that |Oe(Bg/p)| > 8. (9.5) If Bg/p is indecomposable then BG/P is normal by Lemma 8.3 and hence B is CI by Lemma 9.2 and Lemma 5.5. If S has a gwr-complement with respect to BG/P then B is CI by Lemma 9.4. If Oe(BG/P) ^ S then B is CI by Lemma 9.2 and Lemma 5.8. Suppose that none of the above conditions does not hold for B. Then, in view of Equation (9.5), B satisfies all conditions from one of the Cases 1 or 2. Therefore, B is CI. Clearly, AG/L = A(G/P)/A and hence AG/L is normal. Also AG/L is CI by Lemma 9.2. The S-ring B is CI by the above paragraph. Thus, A is CI by Lemma 4.4. Case 4: |Oe(Ag/p)| = 2. Let A = Oe(Ag/p). Clearly, A is the least AG/P-subgroup. If | rad(X)| > 1 for every X G S(AG/P) outside S then A < rad(X) for every X G S(AG/P) outside S and we obtain a contradiction with Equation (9.1). So there exists X G S(Ag/p) outside S with | rad(X)| = 1. From Equation (9.2) it follows that |X | > 1. Lemma 8.8 implies that |X | =4. The number A = |X n Ax| does not depend on x G X by Lemma 2.1. If A = 2 then A < rad(X), a contradiction. Therefore A = 1. (9.6) 292 Ars Math. Contemp. 19 (2020) 189-208 One of the statements of Lemma 8.8 holds for AG/P. If Statement (1) of Lemma 8.8 holds for AG/P then there exists Y e S(AG/P) with |Y| = 16 and | rad(Y)| = 16. Since |S| = 16, we conclude that Y lies outside S and hence Y = (G/P) \ S. This means that S is a gwr-complement to S with respect to AG/P. However, this contradicts Equation (9.1). If Statement (2) of Lemma 8.8 holds for AG/P then | AutG/P (AG/P) | > | AutS (AS) |. So Lemma 9.5 implies that AutG/P(AG/P)S = AutS(AS). Therefore, A is CI by Lemma 9.2 and Lemma 5.1 Suppose that Statement (3) of Lemma 8.8 holds for AG/P, i.e. there exists an Ag/p-subgroup B such that |B| e {2,4} and A(G/P)/B is normal. Let L = n-i(B), where n: G ^ G/P is the canonical epimorphism, and B = V(N, G), where N = Aut(A)G/LG right. We prove that B is a CI-S-ring. As in Case 3, B is the S-wreath product by Lemma 5.4 and B > A by Equations (3.1) and (3.2). So BG/P > AG/P and hence BG/P is a 2-S-ring by Lemma 6.1. Therefore B and U satisfy the conditions of Lemma 9.3. Note that X is a BG/P-set and Equation (9.3) holds because BG/P > AG/P. By the definition of B, every basic set of B is contained in an L-coset and hence every basic set of Bg/p is contained in a B -coset. The set X isa BG/P-set with |X | =4 and | rad(X )| = 1. So there exists X1 e S(BG/P) such that Xi C X and |Xi| e {1,2}. If |Xi| = 1 then Xi C Oe (BG/P). If |Xi| = 2 then Xi isacosetbya BG/P-subgroup Ai of order 2. Clearly, Ai C Og(BG/P). In view of Equation (9.6), we have Ai = A. Thus, in both cases (BG/P) ^ A. Together with Equation (9.3) this implies that O(Bg/p)|> 4. (9.7) If BG/P is indecomposable then BG/P is normal by Lemma 8.3 and hence B is CI by Lemma 9.2 and Lemma 5.5. If S has a gwr-complement with respect to BG/P then B is CI byLemma9.4. If Oe (BG/P) ^ S then B is CI by Lemma 9.2 and Lemma 5.8. Suppose that none of the above conditions does not hold for B. Then, in view of Equation (9.7), B satisfies all conditions from one of the Cases 1, 2 or 3. Therefore, B is CI. The S-ring AG/L is normal because it is isomorphic to A(G/P)/B. The S-rings AG/L and B are CI by Lemma 9.2 and the above paragraph respectively. Thus, A is CI by Lemma 4.4. All cases were considered. □ Proof of Theorem 9.1. Let Hi be a maximal A-subgroup contained in H and Pi the least A-subgroup containing P. Lemma 9.6. If Hi = H then A is a CI-S-ring. Proof. The S-ring AG/H is ap-S-ring over G/H = by Lemma 6.4. So AG/H = ZCp. Clearly, G = HPi. Therefore A = AH * APl by Lemma 7.2. Since H and Pi/(H n Pi) are proper sections of G, the S-rings AH and APl/(HnPl) are CI by Lemma 9.2. Thus, A is CI by Lemma 5.6. □ Lemma 9.7. If Hi < H and HiP^i = G then A is a CI-S-ring. G. Ryabov: The Cayley isomorphism property for the group C| x Cp 293 Proof. Since H1 = (H1P1)p/ = H, Lemma 7.1 implies that A = AHl * APl. The S-rings AHl and APl/(HnPl) are CI by Lemma 9.2 because H1 and P1/(H1 n P1) are proper sections of G. Therefore A is CI by Lemma 5.6. □ In view of Lemma 9.6, we may assume that H1 < H. Then one of the statements of Lemma 7.3 holds for A. If Statement (1) of Lemma 7.3 holds for A then A = Ahi I Aa/Hl, where rk(AG/Hl) = 2. If H1 is trivial then rk(A) = 2. Obviously, A is CI in this case. If H1 is nontrivial then A is CI by Lemma 9.2 and Lemma 5.2. Assume that Statement (2) of Lemma 7.3 holds for A, i.e. A = Au is Ag/p1 , where U = H1P1, S = U/P^ and P1 < G. In view of Lemma 9.7, we may assume that H1P1 < G, i.e. A is the nontrivial S-wreath product. 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