Some new groups which are not CI-groups with respect to graphs

A group G is a CI-group with respect to graphs if two Cayley graphs of G are isomorphic if and only if they are isomorphic by a group automorphism of G. We show that an infinite family of groups which include Dn × F3p are not CI-groups with respect to graphs, where p is prime, n 6= 10 is relatively prime to 3p, Dn is the dihedral group of order n, and F3p is the nonabelian group of order 3p.

The Cayley isomorphism problem has been studied extensively for the last 50 years.This problem is actually two different, but highly related problems.The most general version asks for necessary and sufficient conditions to determine isomorphism between two Cayley (di)graphs of a group G. Usually what is meant by "necessary and sufficient conditions" is for an explicit and minimal list L (which may or may not depend on the specific Cayley (di)graphs under consideration) of elements of S G , and which satisfies the statement two Cayley (di)graphs of G are isomorphic if and only if they are isomorphic by an element on the list L. The less general problem specifies the minimal list L in advance as the group automorphisms of G, and asks for which G this list is necessary and sufficient.The choice of group automorphisms as the minimal list is because the image of a Cayley (di)graph of G under a group automorphism is also a Cayley (di)graph of G, and so one must check whether group automorphisms are (di)graph isomorphisms.
In this paper, we contribute to the second problem by considering an infinite family F of groups which include D n × F 3p , where D n is the dihedral group of order n and F 3p is the electronic journal of combinatorics 25(1) (2018), #P1.12 the nonabelian group of order 3p, with n = 10 relatively prime to 3p and p a prime.We show that testing only the group automorphisms of a group G in F is not sufficient to test for isomorphism among Cayley graphs of G. Using the standard terminology for this problem (defined below), we show these groups are not CI-groups with respect to graphs.We remark that word graph here is chosen deliberately, as this fact is already know for Cayley digraphs of these groups [10].
Definition 1.Let G be a group and S ⊂ G such that 1 ∈ S and S = S −1 = {s −1 : s ∈ S}.Define a Cayley graph of G, denoted Cay(G, S), to be the graph with V (Cay(G, S)) = G and E(Cay(G, S)) = {{g, gs} : g ∈ G, s ∈ S}.We call S the connection set of Cay(G, S).Definition 2. A group G S X , where S X is the symmetric group on the set X, is transitive if whenever x, y ∈ X then there exists g ∈ G with g(x) = y.
and so G L = {g L : g ∈ G} Aut(Cay(G, S)).Here Aut(Cay(G, S)) is the group of all automorphisms of Cay(G, S).The group G L is the left regular representation of G.It is easy to see that G L S G is a transitive group, and so Cayley graphs are vertex-transitive graphs, that is, graphs whose automorphism group is transitive on their vertex-set.More generally, a transitive group H S X is regular if the stabilizer in H of a point is trivial.Equivalently, |H| = |X|.This is where the word "regular" in "left regular representation" comes from.Definition 3. Let G be a group.We say G is a CI-group with respect to graphs if whenever S, T ⊂ G with S −1 = S and T −1 = T , then Cay(G, S) and Cay(G, T ) are isomorphic if and only there exists a group automorphism α ∈ Aut(G) with α(Cay(G, S)) = Cay(G, T ).
It is easy to show that if α ∈ Aut(G), then α(Cay(G, S)) = Cay(G, α(S)).Thus if testing isomorphisms between two Cayley graphs of G, the group automorphisms of G must be checked.The notion of a graphical regular representation or GRR of a group G will be crucial in our construction.All groups which have a GRR are known, see [7].There are two infinite families of groups G which do not have GRR's, namely abelian groups and generalized dicyclic groups (the interested reader is referred to [7] for the definition of a generalized dicyclic group).Additionally, there are 13 groups of small order not in these two infinite families which do not have GRR's, and one of these groups, namely D 10 , will play a role in this paper.We now define some groups which will be of interest in this paper.Definition 5. Let M be an abelian group such that every Sylow p-subgroup of M is elementary abelian.Denote the largest order of any element of M by exp(M ).Let the electronic journal of combinatorics 25(1) (2018), #P1.12 n ∈ {2, 3, 4, 8} be relatively prime to |M |.Set E(n, M ) = Z n φ M , where if n is even then φ(g) = g −1 , while if n = 3 then φ(g) = g , where is an integer satisfying 3 ≡ 1 ( mod exp(M )) and gcd( ( − 1), exp(M )) = 1.
If M = Z p , and 3|(p − 1) then E(3, Z p ) is the nonabelian group of order 3p, which we denote by F 3p (as this group is a Frobenious group).Similarly, E(2, Z n ) is the dihedral group of order 2n.The next result is a combination of results of Li, Lu, and Pálfy [9], and Somlai [11], and lists all possible CI-groups with respect to graphs.Not every group in this result is known to be a CI-group with respect to graphs -see [6] for a recent list of the known CI-groups with respect to graphs.Theorem 6.Let G be a CI-group with respect to graphs.(c) H 3 is isomorphic to one of the groups E(3, M ), A 4 , or 1.
Before turning to our results, we need some additional terms and notation.Intuitively, Γ/B is obtained by identifying all the vertices in each block of B, then eliminating loops and multiple edges.Additionally, if Γ is a vertex-transitive graph with G Aut(Γ) transitive with an invariant partition B, then G/B Aut(Γ/B).We will need the following technical lemma.
Lemma 10.Let G be a group and If R is a CI-group with respect to graphs, then Γ 1 and Γ 2 are isomorphic by and element of S G that normalizes R and fixes g.
Theorem 11.Let p 5 be prime such that 3|(p − 1), G be a group of order relatively prime to 3p that has a GRR whose connection set contains a non self-inverse element, and F 3p be the nonabelian group of order 3p.Then G × F 3p is not a CI-group with respect to graphs.
Towards a contradiction, suppose α ∈ S G×Z 3 ×Zp fixes (1 G , 0, 0), normalizes R, and α(Γ) = Γ .As α normalizes R and B is the unique invariant partition of R with blocks of size 3p, we see B is also an invariant partition of α, R .This follows as α(B) is then an invariant partition of R, and so α(B) = B. Then α/B : Γ/B → Γ /B is an isomorphism that normalizes G L and fixes 1 G , and so by [ p .As α maps the neighbors of (1 G , 0, 0) to the neighbors of (1 G , 0, 0), However, no element of the form (t, 1, j) is contained in ι(S) for any j ∈ Z p , a contradiction.The result follows by Lemma 10.
Corollary 12. Let M be an abelian group that contains an element of prime order p 5.
, either |M | is divisible by a prime q other than 5 or by 25.Note that q = 2 or 3 as |M | is relatively prime to 6. Then there exists L H × F 3p such that (H × F 3p )/L ∼ = D 2q × F 3p , where either q = 25 or q 7 is prime.Again by [5,Theorem 8] it suffices to show that D 2q × F 3p is not a CI-group with respect to graphs.As D 2q has a GRR whose connection set contains an element of order q by the proof of [12,Theorem 2], the result follows in this case by Theorem 11.Noting that if H 3 = A 4 then H 2 = 1 in Theorem 6, combining Corollary 12 with Theorem 6 we have the following improvement to Theorem 6.
Corollary 13.Let G be a CI-group with respect to graphs.(c) H 3 is isomorphic to one of the groups D 10 , or 1.
3. If G contains elements of order 9, then G is one of the groups Z 2 Z 9 , Z 4 Z 9 , Z 9 Z 2 2 , or Z n 2 × Z 9 , with n 5.
We remark that it has been shown that E(3, p) is a CI-group with respect to graphs [2], and some groups H 1 × E(3, M ) with H 1 = 1 as in the above result are CI-groups with respect to graphs [3,Theorem 22].

Definition 4 .
A graphical regular representation or GRR of a group G is a Cayley graph Γ of G such that Aut(Γ) = G L .
is also a block of G for every g ∈ G.The set {g(B) : B ∈ B} is an invariant partition of G. Definition 8. Let G S X be transitive with invariant partition B. An element g ∈ G induces a permutation g/B on B given by g/B(B) = B if and only if g(B) = B .We set G/B = {g/B : g ∈ G}.Definition 9. Let Γ be a vertex-transitive graph and G Aut(Γ) be transitive with invariant partition B. Define the block quotient graph of Γ with respect to B, denoted Γ/B, to be the graph with vertex set B and edge set {{B, B } : B = B ∈ B and uv ∈ E(Γ) for some u ∈ B and v ∈ B }.

1 ,
Corollary 4.2B] is an automorphism of G.As Γ/B ∼ = Γ /B = Cay(G, T ), we see α/B ∈ Aut(Cay(G, T )) = G L , and so α/B = 1.It is clear Ψ and β centralize { hL : h ∈ G} and by [4, Lemma 2.5] Ψ and β normalize ρ, τ .Then Ψ, β ∈ N S K (R).As α/B = 1, we conclude by [4, Lemma 2.5] and [1, Corollary 4.2B] that α ∈ Ψ, β .As Ψ Ψ, β , we may write α = Ψ a βb where a ∈ Z p and b ∈ Z * 1.If G does not contain elements of order 8 or 9, then G = H 1 × H 2 × H 3 , where the orders of H 1 , H 2 , and H 3 are pairwise relatively prime, and (a) H 1 is an abelian group, and each Sylow p-subgroup of H 1 is isomorphic to Z k p for k < 2p + 3 or Z 4 ; (b) H 2 is isomorphic to one of the groups E(2, M ), E(4, M ), Q 8 , or 1; [7], or E(4, M ), where M is an abelian group of order relatively prime to 6 and |M |.If H = D 10 then H × E(3, M ) is not a CI-group with respect to graphs.F 3p .As quotients of CIgroups with respect to graphs are CI-groups with respect to graphs[5, Theorem 8], it suffices to show that H × F 3p is not a CI-group with respect to graphs.The groups Q 8 and E(4, M ) have a GRR[7]which is of course connected.As Q 8 contains a unique subgroup of order 2 and is a 2-group, any generating set of Q 8 must contain an element of order 4. As E(4, M ) contains a unique subgroup H of index 2 that contains the unique element of E(4, M ) of order 2, any generating set of E(4, M ) must also contain an element of order 4. The result follows by Theorem 11 in the cases H Proof.There exists N M such that M/N ∼ = Z p and, as φ (as in the definition of E(3, M )) fixes every subgroup of M , N E(3, M ).Then E(3, M )/N ∼ = 1.If G does not contain elements of order 8 or 9, then G = H 1 × H 2 × H 3 , where the orders of H 1 , H 2 , and H 3 are pairwise relatively prime, and (a) H 1 is an abelian group, and each Sylow p-subgroup of H 1 is isomorphic to Z k p for k < 2p + 3 or Z 4 ; (b) H 2 is isomorphic to one of the groups E(2, M ), E(3, M ), E(4, M ), Q 8 , A 4 , or 1;