ARS MATHEMATICA CONTEMPORANEA
Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 8 (2015) 433-444
The Cayley isomorphism property for groups of order 8p
Gabor Somlai *
EotvOs Lorând University, Department of Algebra and Number Theory, 1117 Pâzmâny Péter sétâny 1/C, Budapest, Hungary
Received 31 December 2013, accepted 10 September 2014, published online 26 June 2015
For every prime p > 3 we prove that Q x zp is a DCI-group, where Q denotes the quaternion group of order 8. Using the same method we reprove the fact that z3 x zp is a CI-group for every prime p > 3, which was obtained in [3]. This result completes the description of CI-groups of order 8p.
Keywords: Cayley graphs, CI-groups. Math. Subj. Class.: 05C25
1 Introduction
Let G be a finite group and S a subset of G. The Cayley graph Cay(G, S) is defined by having the vertex set G and g is adjacent to h if and only if g-1 h G S. The set S is called the connection set of the Cayley graph Cay(G, S). A Cayley graph Cay(G, S) is undirected if and only if S = S-1, where S-1 = { s-1 G G | s G S }. Every left multiplication via elements of G is an automorphism of Cay(G, S), so the automorphism group of every Cayley graph on G contains a regular subgroup isomorphic to G. Moreover, this property characterises the Cayley graphs of G.
Similarly to Cayley graphs one can also define ternary Cayley relational structures. (V, E1, E2,..., El) is a colour ternary relational structure if E C V3 for i = 1,... ,l. We say that a colour ternary relational structure (V, E1,..., El) is a Cayley ternary relational structure of the group G if the automorphism group of (V, E1,..., El) contains a regular subgroup isomorphic to G.
It is clear that Cay(G, S) = Cay(G, S)) for every p G Aut(G). A Cayley graph Cay(G, S) is said to be a Cl-graph if, for each T C G, the Cayley graphs Cay(G, S)
* Research supported by the Hungarian Scientific Fund (OTKA), grant no. K84233 E-mail address: gsomlai@cs.elte.hu (Gabor Somlai)
Abstract
©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
378
Ars Math. Contemp. 8 (2015) 365-434
and Cay(G, T) are isomorphic if and only if there is an automorphism ^ of G such that ^(S) = T. Furthermore, a group G is called a DCI-group if every Cayley graph of G is a Cl-graph and it is called a Cl-group if every undirected Cayley graph of G is a Cl-graph.
Similarly, a group G is called a Cl-group with respect to colour ternary relational structures, if for any pair of isomorphic colour ternary relational structures of G there exists an isomorphism induced by an automorphism of G.
Let G be a Cl-group of order 8p, where p is an odd prime. It is easy to verify that z2 x z4 and the dihedral group of order 8 are not Cl-groups. It can easily be seen that every subgroup of a Cl-group is also a Cl-group. Therefore the Sylow 2-subgroup of G can only be z8, z2 or the quaternion group Q of order 8.
It was proved by Li, Lu and Palfy [5, Theorem 1.2 (b)] that a finite Cl-group of order 8p containing an element of order 8 can only be
H = (a, z | ap = 1, z8 = 1, z-1az = a-1) .
It was also shown in [5, Theorem 1.3] that H is a CI-group, though not a DCI-group. In view of these results, for the rest of the discussion, we assume that the Sylow 2-subgroup of G is isomorphic to Q or z2.
It was proved by Dobson [2] that z2 xZp is a CI-group with respect to ternary relational structures if p > 11. Moreover, Dobson and Spiga [3] proved that z2 x zp is a DCI-group with respect to colour ternary relational structures if and only if p = 3 and 7. As a consequence of this result it was proved in [3] that z3 x zp is a DCI-group for all primes p.
If p > 8 or p = 5, then by Sylow's Theorem the Sylow p-subgroup of G is a normal subgroup, therefore G is isomorphic to one of the following groups: zf x zp, Q x zp, z3 k zp or Q k zp. It can also be seen from [5, Theorem 1.2] that neither Q k zp nor z3 k zp is a CI-group.
If p = 7, then either the Sylow 7-subgroup is normal, in which case G is as before, or G has 8 Sylow 7-subgroups and the Sylow 2-subgroup of G is normal. Then the Sylow 7-subgroup of G acts transitively by conjugation on the the non-identity elements of the Sylow 2-subgroup. Hence G = z3 x z7, which is not a CI-group by [5, Theorem 1.2.(b)].
If p = 3, then the order of G is 24. A complete list of CI-groups of order at most 31 was given in the Ph.D. thesis of Royle, see [7]. The CI-groups of order 24 are the following: Q x z3, z8 k z3 and z3 x z3.
Spiga [6] proved that Q x z3 is not a CI-group with respect to colour ternary relational structures.
Using different methods depending on whether p > 8, or p = 5,7 we show that the other groups are DCI-groups. By extending our result with the fact that z2 x z3 is a CI-group we get that Q x zp is a CI-group for every odd prime p.
Theorem 1.1. For every prime p > 3 the group Q x zp is a DCI-group.
We also prove the following result which was first obtained in [3].
Theorem 1.2 (Dobson, Spiga [3]). For every prime p > 3 the group z2 x zp is a DCI-group.
Our paper is organized as follows. In Section 2 we introduce the notation that will be used throughout this paper. In Section 3 we collect important ideas which are useful in the proof of Theorem 1.1 and 1.2. Section 4 contains the proof of Theorem 1.1 and 1.2 for primes p > 8 and Section 5 contains the proof of Theorem 1.1 and 1.2 for p = 5 and 7.
G. Somlai: The Cayley isomorphism property for groups of order 8p
435
2 Technical details
In this section we introduce some notation. Let G be a group. We use H < G to denote that H is a subgroup of G and by NG(H) and CG(H) we denote the normalizer and the centralizer of H in G, respectively.
Let us assume that the group H acts on the set Q and let G be an arbitrary group. Then by G fa H we denote the wreath product of G and H. Every element g e G fa H can be uniquely written as hk, where k e K = f]uen Gu and h e H. The group K = f]uen Gu is called the base group of G H and the elements of K can be treated as functions from Q to G. If g e G H and g = hk we denote k by (g)b. In order to simplify the notation Q will be omitted if it is clear from the definition of H and we will write GI H.
The symmetric group on the set Q will be denoted by Sym(Q). Let G be a permutation group on the set Q. For a G-invariant partition B of the set Q we use GB to denote the permutation group on B induced by the action of G and similarly, for every g e G we denote by gB the action of g on the partition B.
For a group G, let G denote the subgroup of the symmetric group Sym(G) formed by the elements of G acting by right multiplication on G. For every Cayley graph r = Cay(G, S) the subgroup G of Sym(G) is contained in Aut(r).
Definition 2.1. Let G < Sym(Q) be a permutation group. Let
G(2) = n G Sym(Q)
Va, b G Q 3ga,b G G with n(a) = ga,b(a) and
n(b) = ga,b(b)
We say that G(2) is the 2-closure of the permutation group G.
The following lemma is well-known and follows directly from the definition of G(2). Lemma 2.2. Let r be a graph. If G < Aut(r), then G(2) < Aut(r).
3 Basic ideas
In this section we collect some results and some important ideas that we will use in the proof of Theorem 1.1 and Theorem 1.2.
We begin with a fundamental lemma that we will use all along this paper.
Lemma 3.1 (Babai [1]). The Cayley graph Cay(G, S ) is a CI-graph if and only if for every regular subgroup G of Aut(Cay(G, S)) isomorphic to G there is a p G Aut(Cay(G, S)) such that Gm = G.
We introduce the following definition.
Definition 3.2. (a) We say that a Cayley graph Cay(G, S) is a ci(2) -graph if and only if for every regular subgroup G of Aut(Cay(G, S)) isomorphic to G there is a a G {G, G)(2) such that Gff = G.
(b) A group G is called a DCI(2)-group if for every S c G the Cayley graph Cay(G, S) is aCI(2) -graph.
Let R be either Q or z3. Let us assume that A = Aut(Cay(G, S)) < Sym(8p) contains two copies of regular subgroups, R x Zp and R x zp. By Sylow's theorem we may assume that zp and zp are in the same Sylow p-subgroup P of Sym(8p). If p > 8,
378	Ars Math. Contemp. 8 (2015) 365-436
then P is isomorphic to zp. Moreover, P is generated by 8 disjoint p-cycles. It follows that both R and R normalize P so we may assume that R and R lie in the same Sylow 2-subgroup of NA(P). Let P2 denote a Sylow 2-subgroup of Sym(8). It is also well known that P2 is isomorphic to the automorphism group of the following graph A:
1 2	3 4 5 6	7 8
Every automorphism of A permutes the leaves of the graph and the permutation of the
leaves determines the automorphism, therefore Aut(A) can naturally be embedded into
Sym(8).
Lemma 3.3. (a) There are exactly two regular subgroups of P2 which are isomorphic to Q.
(b) There are exactly two regular subgroups of P2 which are isomorphic to Zf.
Proof. (a) Let Q be a regular subgroup of Aut(A) isomorphic to the quaternion group with generators i and j. Since Q is regular, for every 1 < m < 4 there is a qm e Q (not necessarily distinct) such that qm (2m - 1) = 2m. These are automorphisms of A so qm(2m) = 2m - 1 and hence since Q is regular the order of qm is 2. There is only one involution in Q so qm = i2 for every 1 < m < 4 and this fact determines completely the action of i2 on A. Note that the automorphisms qm are all equal.
We can assume that i(1) = 3. Such an isomorphism of A fixes setwise {1,2,3,4} so we have that i(3) = 2, i(2) = 4 and i(4) = 1 since i is of order 4. Using again the fact that Q is regular on A and i2 (5) = 6, we get that there are two choices for the action of i: i = (1324)(5768) or i = (1324)(5867).
We can also assume that j(1) = 5. This implies that j(5) = j2(1) = i2(1) = 2, and j(2) = 6 since j e Aut(A) and j(6) = 1. The action of i determines the action of j on A since iji = j. Applying this to the leaf 3 we get that j(3) = 8 if i = (1324)(5768) and j(3) = 7 if i = (1324)(5867) so there is no more choice for the action of j. Finally, i and j generate Q and this gives the result.
(b) One can prove this using an argument similar to the previous case.
□
The previous proof also gives the following.
G. Somlai: The Cayley isomorphism property for groups of order 8p
437
Lemma 3.4. (a) The following two pairs of permutations generate the two regular subgroups of Aut(A) < Sym(8) isomorphic to Q:
11	= (1324)(5768), j1 = (1526)(3748) and
12	= (1324)(5867), j2 = (1526)(3847).
(b) The elements of these regular subgroups of Aut(A) are the following:
Qi:		Qr •	
id	(12)(34)(56)(78)	id	(12)(34)(56)(78)
(1324)(5768)	(1423)(5867)	(1324)(5867)	(1423)(5768)
(1526)(3847)	(1625)(3748)	(1526)(3748)	(1625)(3847)
(1728)(3546)	(1827)(3645)	(1728)(3645)	(1827)(3546)
Using the identification given in the following table, Qi and Qr act on Q by left-and right-multiplication with the elements of Q, respectively:
Q
1	2	3	4	5	6	7
1	-1	i	—i	j		k
8 -k
(c) A1 = (x1,x2,x3) and A2 = isomorphic to Z2, where
xi = (12)(34)(56)(78), x2 yi = (12)(34)(56)(78), y2 =
(y1,y2,y3) are subgroups of Aut(A) < Sym(8)
= (13)(24)(57)(68), x3 = (15)(26)(37)(48) and = (13)(24)(58)(67), y3 = (15)(26)(38)(47).
Lemma 3.5. Let us assume that G1 < P2 is generated by two different regular subgroups Qa and Qb of Aut(A) which are isomorphic to Q and G2 < P2 is generated by two different regular subgroups A1 and A2 of Aut(A) which are isomorphic to Zf. Then
Gi = G2 .
Proof. It is clear that |P21 = | Aut(A) | = 27. One can see using Lemma 3.4 (a) and (c) that G1 and G2 are generated by even permutations. Both G1 and G2 induce an action on the set V = {A, B, C, D} which is a set of vertices of A and it is easy to verify that every permutation of V induced by G1 and G2 is even. This shows that G1 and G2 are contained in a subgroup of P2 of cardinality 25.
Lemma 3.4 (b) shows that |Qa n Qb| = 2 and one can also check using Lemma 3.4 (c) that |A1 n A2| = 2. This gives |G1| > 25 and |G21 > 25, finishing the proof of Lemma 3.5.	□
Proposition 3.6. (a) The quaternion group Q is a DC I (2)-group. (b) The elementary abelian group Z2 is a DC I (2)-group.
Proof. (a) Let Qa and Qb be two regular subgroups of Sym(8) isomorphic to the quaternion group Q. By Sylow's theorem we may assume that Qa and Qb lie in the same Sylow 2-subgroup of H = (Qa, Qb). Since every Sylow 2-subgroup of H is contained in a Sylow 2-subgroup of Sym(8), we may assume that Qa and Qb are subgroups of Aut (A).
378
Ars Math. Contemp. 8 (2015) 365-438
Our aim is to find an element n G (Qa, Qb}(2) suchthat Qna = Qb. Let us assume that Qa = Qb. Using Lemma 3.4 (a) we may also assume that Qa and Qb are generated by the permutations (1324)(5768), (1526)(3748) and (1324)(5867), (1526)(3847), respectively. Lemma 3.4 (b) shows that H contains the following three permutations:
(12)(34) = (1324)(5768)(1324)(5867) (12)(56) = (1526)(3748)(1526)(3847) (12)(78) = (1728)(3546)(1728)(3645).
Now one can easily see that the permutation (12) is in Finally, it is also easy to check using Lemma 3.4 (b) that Qai2) = Qb.
(b) One can prove this statement using Lemma 3.4 and Lemma 3.5.
Definition 3.7. Let r be an arbitrary graph and A, B c V(r) such that A n B = 0. We write A ~ B if one of the following four possibilities holds:
(a)	For every a G A and b G B there is an edge from a to b but there is no edge from b to a.
(b)	For every a G A and b G B there is an edge from b to a but there is no edge from a to b.
(c)	For every a G A and b G B the vertices a and b are connected with an undirected edge.
(d)	There is no edge between A and B.
We also write A ^ B if none of the previous four possibilities holds.
The following lemma follows easily:
Lemma 3.8. Let A, B be two disjoint subsets of cardinality p of a graph. We write Au B = zp U zp. Let us assume that a generator g of zp acts by g(a1,a2) = (a1 + 1, a2 + 1) on A U B and for a generator a of the cyclic group Zp the action of a is defined by a(a1, a2) = (a1 + b, a2 + c) for some b,c G zp.
(a)	If b = c, then the action of Zp and Zp on A U B are the same.
(b)	If A ^ B, then b = c.
(c)	If A ~ B, then every n G Sym(A U B) which fixes A and B setwise is an automorphism of the graph defined on A U B as long as n| A G Aut(A) and n|B G Aut(B).
4 Main result for p > 8
In this section, we will
prove that R x zp is a DCI-group if p > 8, where R is either Q or
z3.
Proposition 4.1. For every prime p > 8, the group R x Zp is a DCI-group.
Our technique is based on Lemma 3.1 so we have to fix a Cayley graph r = Cay(R x zp,S). Let A = Aut(r) and G = R xZp be aregular subgroup of A isomorphic to Rxzp. In order to prove Proposition 4.1 we have to find an a G A such that Ga = G = R x zp. We will achieve this in three steps.
G. Somlai: The Cayley isomorphism property for groups of order 8p
439
4.1	Step 1
Since p > 8, the Sylow p-subgroup of Sym(8p) is generated by 8 disjoint p-cycles. We may assume zp and zp lie in the same Sylow p-subgroup P of Sym(8p). Then both R and R are subgroups of NSym(8p) (P) n A so we may assume that R and R lie in the same Sylow 2-subgroup of NSym(8p) (P) n A which is contained in a Sylow 2-subgroup of A.
Since p > 8, the Sylow p-subgroup P gives a partition B = {B^ B2,..., B8} of the vertices of r, where |Bj | = p for every i = 1,..., 8 and B is P-invariant. It is easy to see that B is invariant under the action of R and R and hence (G, G) < Sym(p) l Sym(8). Moreover, both G and G are regular so R and R induce regular action on B which we denote by R1 and R2, respectively. The assumption that R and R lie in the same Sylow 2-subgroup of A implies that R1 and R2 are in the same Sylow 2-subgroup of Sym(8).
4.2	Step 2
Let us assume that R1 = R2. We intend to find an element a e A such that (Ra) = R2.
We define a graph r0 on B such that Bm is adjacent to Bn if and only if Bm ^ Bn. This is an undirected graph with vertex set B and both R1 and R2 are regular subgroups of Aut(r0). It follows that r0 is a Cayley graph of R.
Observation 4.1. Since R1 < Aut(r0) acts transitively on B we have that the order of eah connected component of r0 divides 8.
We can also define a coloured graph r1 on B by colouring the edges of the complete directed graph on 8 vertices. The vertex Bm is adjacent to the vertex Bn with the same coloured edge as Bm> is adjacent to Bn> in r1 if and only if there exists a graph isomorphism ^ from the induced subgraph of r on Bm U Bn to the induced subgraph of r on Bm/ U B„/ such that ^(Bm) = Bm> and ^(B„) = Bn. The graph r1 is a coloured Cayley graph of R. Moreover, both R1 and R2 act regularly on r1. Using the fact that R has property DC/(2), it is clear that there exists an a' e (R1, R2)(2) < Aut(r1) such that R2 = R1. We would like to lift a' to an automorphism a of r such that aB = a'.
(a) Let us assume first that r0 is a connected graph.
Lemma 4.2. (a) R x zp < zp l Sym(8). (b) If R x zp < zp l Sym(8), then for every r e R we have ( r)b = id.
Proof. (a) We first prove that zp = zp. Let x and y generate zp and zp, respectively. Since x and y lie in the same Sylow p-subgroup and |B1| = p, we can assume that x|Bl = y|Bl. Using Lemma 3.8(b) we get that x|Bm = y|Bn if there exists a path in r0 from Bm to Bn. This shows that x = y since r0 is connected. Moreover, R x zp < zp l Sym(8) since the elements of zp and the elements of R commute. (b) Let A' = A n ^zp l Sym(8^. We have already assumed that R and R lie in
the same Sylow 2-subgroup of A'. Let r be an arbitrary element of R . For every (a, u) e R x zp we have r(a, u) = (b, u + t) for some b e R and t e zp, where t only depends on r and a since r < zp l Sym(8). The permutation group G is transitive, hence there exist r1, r2 e R such that r1(1, u) = (a, u)
378
Ars Math. Contemp. 8 (2015) 365-440
and r2 (b, u +t) = (1, u +t). The order of f2rfi is a power of 2 since r2, r, f lie in a Sylow 2-subgroup. Therefore t = 0 and hence (r)b = id.
□
Lemma 4.2 says that if r0 is connected, then {R, R) < Zp I Sym(8) and (r)b = id for every r G {R, R). Therefore we can define a = a'idB to be an element of the wreath product zp I Sym(8) and clearly a'idB is an element of A with aB = a'.
(b)	Let us assume that r0 is the empty graph.
Then Lemma 3.8(c) shows that every permutation in {R1, R2)(2) lifts to an automorphism of r.
(c)	Let us assume that r0 is neither connected nor the empty graph. Observation 4.2. If R1 = R2, then {R, R) < A contains #2, 33 such that
= (BiB2)(BsB4), 3B = (BiB2)(B5B6), ySf = (B^XBrBs).
Proof. Recall from Lemma 3.5 that {R, R) is the same group whether R is Q or zf. By Lemma 3.4 the elements #2, 33 can be generated as products of an element of R and R, as in the proof of Proposition 3.6, if R = Q.	□
Lemma 4.3. We claim that B2k-1 and B2k are in the same connected component of r0 for k = 1,2,3,4.
Proof. Since r0 is a Cayley graph and R1 is transitive on the pairs of the form (B2k_1, B2k) it is enough to prove that B1 and B2 are in the same connected component of r0. If B1 ^ B2, then B1 is adjacent to B2 in r0, so we can assume that B1 ~ B2. Since r0 is not the empty graph B1 is adjacent to Bi for some l > 2, so B1 ^ Bi. By Observation (4.2) there exists 3 G A such that ^(B1) = B2 and ^(Bi) = Bi. This shows that B2 ^ Bi and hence there is a path from B1 to B2 in
ro.	□
r0 is not connected, so the order of the connected components of r cannot be bigger than 4. Since B1 and B2 are in the same connected component of r0 there exists a partition H1 U H2 = B such that |H11 = |H21 =4, B1, B2 G H1 and no vertex in H1 is adjacent to any vertex of H2 in r0.
Lemma 4.4. There exists a G A such that aB = a'.
Proof. Let us assume first that H1 = {B1, B2, B3, B4}. Then we define a1 to be equal to 32 on H1 and the identity on H2, where 32 is defined in Observation 4.2. Using Lemma 3.8(c) we get that a1 is in {R, R)(2).
If H1 = {B1,Bf,B5,B6} or H1 = {B1,B2,B7,Bg}, then we define af by a2|Hl = and a2|H2 = id, where is defined in Observation 4.2. Lemma 3.8(c) shows again that a2 G A.
It is easy to see that aB — aB — (B1B2). Therefore A contains an element a such
3
that Rf = R2.	□
We conclude that we can assume that R1 = R2.
G. Somlai: The Cayley isomorphism property for groups of order 8p
441
4.3 Step 3
Let us now assume that R1 = R2. We intend to find f G A such that R7 = R.
Let X and X denote the generators of zp and zp, respectively. We may assume that
x|bi = X|Bl .
Lemma 4.5. There exists 7 G A such that XY = X.
Proof. Let us assume first that r0 is connected. It is clear by Lemma 3.8 (b) that X = X. So, we may take 7 = 1.
Let us assume that r0 is not connected. In this case there are at least two connected components which we denote by C,..., Cn. We may assume that B1 G Ci. The permutations X and X are elements of the base group of zp I Sym(8) and hence they can be considered as functions on B. We may assume that X(r, u) = (r, u + 1) for every (r, u) G R x zp. By Lemma 3.8 (b), the function X is constant on each equivalence class.
For every 1 < m < n there exists rm G R such that rm (C1 ) = Cm and for every rm G R there exists rm G R such that r^ = . Let 7 be defined as follows:
Y |uCi = id
Y|uCm = rmf-1 for 2 < m < n.
Let (b, v) G rm(Be) with Be G C1 and we denote rm1(6, v) by (a, u). Since X is constant on Cm we have Xs (b, v) = (b, v + cms) for some cm which only depends on Cm. Thus rm(a, u + s) = (b, v + cms) since X and rm commute and X|Be = X|Be. Therefore we have
Y (b, w) = rm(a, w) = rm(a, u + (w - u)) = (b, v + cm(w - u))
for every (b, w) G rm(Be). It is easy to verify that Y-1(b, w) = (b, w-ve+"Cm ) for every w G zp which gives
Y-1Xy(b, w) = y-1 X(b, wcm + v - ucm) = Y-1(b, wcm + v - ucm + cm) = (b, w + 1).
It follows that y-1xy = f.
It remains to show that f G A. Let y and z be two elements of R x zp. We denote by By and Bz the elements of B containing y and z, respectively. If By and Bz are in the same connected component of r0, then either f is defined on By and Bz by rm^1 which is the element of the group (G , G?} < A or f (y) = y and f(z) = z.
If By and Bz are not in the same connected component, then By ~ Bz. The definition of f shows that fB = id. Using Lemma 3.8 (c) we get that f|By uBz is an automorphism of the induced subgraph of r on the set By U Bz, which proves that f G A, finishing the proof of Lemma 4.5.	□
Using Lemma 4.5 we may assume that X = X. Since X and r commute we have R x Zp < zp I Sym(8). Now we can apply Lemma 4.2 which gives ( r)b = id for every r G R. This proves that R = R since R1 = R2. Therefore G = G, finishing the proof of Proposition 4.1.	□
It is straightforward to check that the proof of Proposition 4.1 only uses the fact that p > 8 in the first step of the argument. We can formulate this fact in Proposition 4.6.
378
Ars Math. Contemp. 8 (2015) 365-442
Proposition 4.6. Let r be a Cayley graph of G = Q x zp or G = z3 x zp, where p is an odd prime and let G = Q x Zp or G = z3 x Zp be a regular subgroup of Aut(r) isomorphic to G. Let us assume that there exists a (G, G)-invariantpartition B = {Bi, B2,..., B8} of V(r), where B | = p for every i = {1,..., 8}. In addition, we assume that Zp is a subgroup of the base group of zp I Sym(B). Then there is an automorphism a of the graph r such that Ga = G.
5 Main result for p = 5 and 7
In this section we will prove that Q x z5, Q x z7, zf x z5 and zf x z7 are CI-groups.
The whole section is based on the paper [5], so we will only modify the proof of Lemma 5.4 of [5].
Proposition 5.1. Every Cayley graph of Q x z5, Q x z7, z2 x z5 and z3 x z7 is a Cl-graph.
We let R denote either Q or z3, and let p = 5 or 7. Let r be a Cayley graph of R x Zp and let A = Aut(r). We denote by P a Sylow p-subgroup of A. Let us assume that A contains two copies of regular subgroups which we denote by G = R x zp and G = R x zp. We can assume that r is neither the empty nor the complete graph and both zp and zp are contained in P.
If the order of every orbit of P on V(r) is p, then it is clear from Proposition 4.6 that r is a CI-graph. Therefore P has an orbit A c G such that |A| = p2 since p3 > |G|. The remaining orbits of P have order p since 2p2 > 8p.
It was proved in [5] Lemma 5.4 that the action of A on the vertices of the graph r cannot be primitive so there is a nontrivial A-invariant partition B = {B0, B1,..., Bi-1} of V(r) = G. The elements of the partition B have the same cardinality since the action of A is transitive on B so |Bj| < 4p < p2 for every i = 0,1,..., t — 1. The partition B is P-invariant so P acts on B. Since P is a p-group, the order of every orbit of P is a power of p.
Let C = {Co, Ci,..., Cs_i} be an orbit of P on B such that A C uS-dCj. We may assume that Bj = Cj for i = 0,1,..., s — 1. It is clear that s is a power of p. If s > p2, then |u|=T01Ci| > 2p2 > 8p which is a contradiction. Since |C0| = |B0| < p2, we cannot have s = 1 . It follows that 1 < s < p2 which implies s = p.
For every i < s and every x e P the following eqalities hold for some j < s
x(Bj n A) = x(Bj) n x(A) = Bj n A.
This implies that
|Bo n A| = |Bj n A|
for every 0 < i < s. Therefore
p2 = | A | = lus-o1 (Bj n A) | = s |Bo n A| = p |Bo n A|.
This gives |B0 n A| = p so |B0| can only be p or 8 since |B0| t = 8p and both |B0| and t > s are at least p.
If |B01 = p, then A is the union of p elements of the A-invariant partition B and every orbit A' of P is an element of the partition B if A' = A. For every orbit A' = A of P and
G. Somlai: The Cayley isomorphism property for groups of order 8p
443
for every y G zp U zp we have y(A') = A'. In particular, y(B7) = B7. By Proposition 4.6 we may assume that there exists an element x' in zp U zp such that x'(B0) = Bj for some j = 0, 7 and clearly x'(B7) = B7. Since both G and G are regular there exists a G CA(x') such that a(B0) = B7. Since a and x' commute we have a(Bj) = B7, which contradicts the fact that a(B0) = B7.
We must therefore have |B01 = 8. Let x and x generate zp and zp, respectively. Since G and G are regular we have that neither xB nor xB is the identity, so both x and x are regular on B. Since both xB and xB generate a transitive subgroup in Sym(B) of prime order p > 2, and every for r G R U R the permutation rB commutes with one of these two elements, we have rB = id. Since x and x are in the same Sylow p-subgroup of P we may assume that x(Bj) = x(Bj) = Bi+1 for i = 0,1,... ,p - 1, where the indices are taken modulo p. By Proposition 4.6 we may also assume that x = x.
For every m there exists an l such that the action of xlx-1 is nontrivial on Bm since x = x. Therefore ABm \Bm contains a regular subgroup and a cycle of length p such that p > . A theorem of Jordan on primitive permutation groups, which can also be found in [8, Theorem 13.1.], says that such a permutation group is 2-transitive and hence the induced subgraph of r on Bm is the complete or the empty graph for every m.
Lemma 5.2. Bm ~ Bn for 0 < m < n < p — 1.
Proof. There exists a unique element g G zp < P such that g(Bm) = Bn. We also have a unique element g G zp < P with gB = gB. Since zp is a cyclic group of prime order and x = x we have g = g. Moreover, we may also assume that g\Bm = g\Bm since g = g and the induced subgraphs of r on Bm+c U Bn+c are all isomorphic, where both m + c and n + c are taken modulo p.
Clearly, g = gg-1 isacycle oflength p on Bn. The vertices of V (r) \ A are contained in P-orbits of order p that contain the orbit of the vertex under x, so meet each Bj in a single vertex, so g fixes every vertex of the set Bm U Bn \ A since gB = id.
Let u G Bm \ A. It is enough to show that if u is adjacent to some v G Bn, then u is adjacent to every vertex of Bn. We will prove that A is transitive on the following pairs: {(u, w) \ w G Bn}.
A is transitive on {(u, w) \ w G Bn n supp(g)} = {(u, w) \ w G Bn n A} since g fixes u. Therefore we may assume that v G Bn \ A and we only have to find an element a G A such that a(u) = u and a(v) G Bn n A.
The restriction of g to Bn is a cycle of length p so g does not commute with r \ Bn, where r is an involution of P. Since r and g commute we have that there is a u' G Bm such that rg(u') = gr(u'). Since the action of R is transitive on Bm there exists r G R such that r(u) = u'. Then
( rr) g(u) = rgr(u) = rg(u') = gr(u') = g ( rr) (u) so there exists a' G A such that
a'g(u) = ga'(u).	(5.1)
Let us suppose that v = g(u). Notice that g(u) is in a P-orbit of order p, so g(u) G A. Then the inequality (5.1) gives a'(v) = ga'(u). Since -R\Bm is regular on Bm there exists s G R such that s(u) = a'(u) and since s and g commute we have s(v) = sg(u) = gs(u) = ga'(u). Therefore s(v) = a'(v) and hence s-1 a' fixes u and 5-1a'(v) = v so we may assume that v = g(u).
378
Ars Math. Contemp. 8 (2015) 365-444
If p = 7, then v e Bn n A.
Let us assume that p = 5. We claim that there exists t e R such that t(u) e Bm \ A =
Bm \ supp(g) while t(v) e Bn n A = Bn n supp(g). It is clear that i(Bm n supp(g)) = Bn n supp(g) and g commutes with each element of R. Therefore it is enough to show that if u,v e Bm \ supp(g) with u = v, then there exists t e R such that t(u) e Bm \ supp(g) and t(v) e Bm n supp(g). This can easily be seen from the fact that gcd(|R|, 5) = 1.
The permutations t-1glt fix the vertex u for every 0 < l < 4 and t-1gl1 t(v) = t-1 gl2 t(v) if l1 ^ l2 (modp). At least one of the the four elements t-1gt, t-1g2i, t-1g3t, t-1 g4t of A fixes u and maps v to an element of Bn n supp(g) = Bn n A since |Bn \ supp(g) | = 3, finishing the proof of the fact that Bm ~ Bn for 0 < m = n < 7.	□
Every permutation of V(r) which fixes Bm setwise for every m is an automorphism of r so there is an a e A such that xa = x. Applying Proposition 4.6 we get that there exists a e A such that ^R x = R x Zp, finishing the proof of Proposition 5.1.
References
[1]	L. Babai, Isomorphism problem for a class of point-symmetric structures, Acta Math. Acad. Sci. Hungar. 29 (1977), 329-336.
[2]	E. Dobson, The isomorphism problem for Cayley ternary relational structures for some abelian groups of order 8p, Dicrete Math. 310 (2010), 2895-2909.
[3]	E. Dobson, P. Spiga, Cl-groups with respect to ternary relational structures: new examples, Ars Math. Contemp. 6 (2012), 351-364.
[4]	C. H. Li, On isomorphisms of finite Cayley graphs- a survey, Discrete Mathematics 256 (2002), 301-334.
[5]	C. H. Li, Z. P. Lu, P. P. Palfy, Further restrictions on the structure of finite Cl-groups, J. Algebr. Comb. (2007), 161-181.
[6]	P. Spiga, On the Cayley isomorphism problem for a digraph with 24 vertices, Ars. Math. Contemp. 1 (2008), 38-43.
[7]	G. Royle, Constructive enumeration of graphs, Thesis submitted to The University of Western Australia, 1987.
[8]	H. Wielandt, Finite permutation groups, Academic Press, London-New York, 1964.