On the Cayley isomorphism problem for Cayley objects of nilpotent groups of some orders

We give a necessary condition to reduce the Cayley isomorphism problem for Cayley objects of a nilpotent or abelian group G whose order satisfies certain arithmetic properties to the Cayley isomorphism problem of Cayley objects of the Sylow subgroups of G in the case of nilpotent groups, and in the case of abelian groups to certain natural subgroups. As an application of this result, we show that Zq×Zp×Zm is a CI-group with respect to digraphs, where q and p are primes with p2 < q and m is a square-free integer satisfying certain arithmetic conditions (but there are no other restrictions on q and p).


Introduction
In 1967 Ádám [1] conjectured that any two circulant graphs of order n are isomorphic if and only if they are isomorphic by a group automorphism of Z n .While Ádám's conjecture was quickly shown to be false [4], the conjecture nonetheless generated much interest in the following question: Are two Cayley graphs of a group G isomorphic if and only if they are isomorphic by a group automorphism of G? If so, we say that G is a CIgroup with respect to graphs.This problem naturally generalizes to any class of combinatorial objects (see [11] for several equivalent formulations of the precise definition of a combinatorial object).Namely, is it true that two Cayley objects of a group G in some class K of combinatorial objects are isomorphic if and only if they are isomorphic by a group automorphism of G? If so, we say that G is a CI-group with respect to K. If G is a CI-group with respect to every class of combinatorial objects, we say that G is a CI-group.In 1987, Pálfy [13] proved the following remarkable result: Theorem 1.A group G is a CI-group if and only if gcd(n, ϕ(n)) = 1 or n = 4, where ϕ is Euler's phi function.While Pálfy's result is quite powerful, it does not tell us anything in general about isomorphisms between Cayley objects of a group G if G is not a CI-group, other than there exists isomorphic Cayley objects of G which are not isomorphic by a group automorphism of G.For such groups, we are then left with the question of if two Cayley objects of G are isomorphic, then what are the possible isomorphisms between them?This is sometimes known as the Cayley isomorphism problem.Usually, one would like the solution to this question to be a (hopefully) short list L of possible isomorphisms.That is, two Cayley objects of G are isomorphic if and only if they are isomorphic by a function in the list L. In 1999, Muzychuk [11] showed that if G is a cyclic group of order n and for any distinct primes p and q dividing n we have that q does not divide p − 1, then any two Cayley objects of G are isomorphic by an automorphism that can be found in a natural way from isomorphisms of Cayley objects of prime-power orders that divide n.Thus Muzychuk reduced the Cayley isomorphism problem for Cayley objects of cyclic groups of some orders to the Cayley isomorphism problem for Cayley objects of cyclic groups of prime-power orders.In 2003, the author [3], found a sufficient condition to extend Muzychuk's result to all abelian groups (with the same order conditions), and showed this sufficient condition was satisfied by some abelian groups.In this paper, we extend the author's earlier result to nilpotent groups, as well as to abelian groups with more general order conditions (Theorem 14).Finally, as an application we will extend the list of CI-groups with respect to digraphs by showing that Z q × Z 2 p × Z m is a CI-group with respect to digraphs, where p and q are distinct primes with p 2 < q and m satisfies certain arithmetic conditions (Theorem 31).
Throughout this paper, G is a finite group.For group theoretic terms not defined in this paper, see [2].We begin with some definitions.Definition 2. Let G be a transitive group acting on Ω.Let X be the set of all complete block systems of G. Define a partial order on X by B C if and only if every block of C is a union of blocks of B. We define B| C to be the complete block system of Stab G (C) = {g ∈ G : g(C) = C}, the set-wise stabilizer of C ∈ C, consisting of all those blocks of B that are contained in C, C ∈ C, and remark that B| C is a complete block system of Stab G (C) in its action on C. By fix G (B) we mean the subgroup of G which fixes each block of B set-wise.That is, fix Definition 3. Let n = Π r i=1 p a i i be the prime factorization of n and define Ω : N → N by Ω(n) = Σ r i=1 a i .Setting m = Ω(n), we say a transitive group G of degree n is m-step imprimitive if there exists a sequence of complete block system B 0 ≺ B 1 ≺ . . .≺ B m .Note that B 0 consists of singletons, while B m consists of the entire set on which G acts.A complete block system B will be said to be normal if B is formed by the orbits of a normal subgroup.We will say that G is normally m-step imprimitive if each B i , 0 i m, is formed by the orbits of a normal subgroup of G.

The main tool
In this section, we will give a sufficient condition that will imply that the Cayley isomorphism problem for nilpotent groups of certain orders can be reduced to the Cayley isomorphism problem for groups of prime-power order (Theorem 14 and Corollary 15), and for abelian groups with more general arithmetic conditions (Theorem 14).That these results have implications for the Cayley isomorphism problem is established in Theorem 26.The following result is straightforward, and so its proof is omitted.Lemma 4. Let G 1 , G 2 S n be transitive such that both G 1 and G 2 admit B as a complete block system.Then G 1 , G 2 admits B as a complete block system.
The following result is trivial after observing that the hypothesis implies that fix We will use the following basic (and known) result implicitly throughout the paper.Lemma 6.Let G S n be transitive with H G a transitive abelian subgroup.Then every complete block system of G is normal and is formed by the orbits of a normal subgroup of H.
Proof.Let B be a complete block system of G consisting of m blocks of size k.As a transitive abelian group is regular [14,  Let G be a permutation group acting on X and H a permutation group acting on Y .Define the wreath product of G and H, denoted G H, to be the group of all permutations of G × H of the form (x, y) → (g(x), h x (y)).
the electronic journal of combinatorics 21(3) (2014), #P3.8 Lemma 8. Let n be a positive integer and G 1 , G 2 be transitive abelian groups of degree n such that G 1 , G 2 is m-step imprimitive.Let n = p a 1 1 p a 2 2 • • • p ar r be the prime-power decomposition of n.Then there exists δ ∈ G 1 , G 2 and a sequence of primes q 1 , . . ., q m such that n = q Proof.We proceed by induction on m.If m = 1, then n is prime, and both G 1 and G 2 are Sylow n-subgroups of S n .Hence there exists δ ∈ G 1 , G 2 such that δ −1 G 2 δ = G 1 , and the result is trivial as G 1 , δ −1 G 2 δ is cyclic of order n.Now assume that the result is true for all m − 1 1, and let G 1 , G 2 be transitive abelian groups of degree n, where As G 1 , G 2 is m-step imprimitive, G 1 , G 2 admits a normal complete block system B consisting of n/q m blocks of size q m for some prime q m |n, and both G 1 /B and G 2 /B are transitive abelian groups of degree n/q m and Ω(n/q m ) = m − 1.Furthermore, as [3,Lemma 8], so by the induction hypothesis, there exists for some sequence of primes q 1 , . . ., q m−1 such that n/q m = q 1 • • • q m−1 , and if G 1 , G 2 is solvable, we may take δ 1 = 1.Furthermore, fix G 1 (B) is semiregular of order q m , and fix δ −1 1 G 2 δ 1 (B) is also semiregular of order q m .Hence there exists subgroup, so we may take δ 2 = 1.Let δ = δ 1 δ 2 .As a Sylow q m -subgroup of fix G,δ −1 Gδ (B) is contained in 1 S n/qm Z qm we have that both G 1 and δ −1 G 2 δ normalize 1 S n/qm Z qm .This then implies that Stab G 1 ,δ −1 G 2 δ (B)| B has a normal Sylow q m -subgroup, so that Stab G 1 ,δ −1 G 2 δ (B)| B is permutation isomorphic to a subgroup of AGL(1, q m ) for every B ∈ B. It then follows by the Embedding Theorem [9, Theorem 2.6], that G 1 , δ −1 G 2 δ is permutation isomorphic to a subgroup of AGL(1, q 1 ) (AGL(1, q 2 ) (• • • AGL(1, q m ))), and the result follows by induction.Definition 9. Let π be a set of primes.A π-group G is a group such that every prime By π , we denote the set of primes dividing |G| that are not contained in π.
We shall have need a consequence of the preceding result.
Lemma 10.Let n be a positive integer and π be the set of distinct prime numbers di- r be the prime-power decomposition of n.By Lemma 8, there exists δ 1 ∈ G 1 , G 2 and a sequence q 1 , . . ., q m of primes such that n = q Also by Hall's Theorem, there exists , and H 1 is a solvable π-group.
Let L = L G be the set of all normal complete block systems of a transitive group G. Then is a canonical partial order on L. Define operations ∪ and ∩ on L by B ∪ C is the normal complete block system of G formed by the orbits of fix as both of these groups are normal), and B ∩ C is the normal complete block system of G formed by the orbits of fix G (B) ∩ fix G (C).Notice that both of these operation do in fact give normal complete block systems as fix [6] for terms regarding lattices not defined here.We also have that . By Lemma 6, there exists A, B, C H such that A is formed by the orbits of A, B is formed by the orbits of B, and C is formed by the orbits of C. As B A, we have that B A.
We remark that the previous result is contained in [12,Theorem 2.10] In the following three results, we will have in the hypothesis that gcd(n i , n j •ϕ(n j )) = 1.Notice that this implies that gcd(n i , n j ) = 1, and that if p i |n i is prime, then p i does not divide p j − 1 for any prime p j |n j .
Lemma 12. Let n 1 , . . ., n r be positive integers such that if i = j, then gcd(n i , n j •ϕ(n j )) = 1, π i the set of primes dividing n i , and H i be a transitive, solvable, π i -group of degree n i , Proof.It is not difficult to see that Π r i=1 H i admits complete block systems C i consisting of n/n i blocks of size n i , 1 i r, formed by the orbits of H i .As each H i is solvable, we have that G is solvable, and so contains an The result now follows.
We now need only one more tool to prove the main result (Theorem 14) of this section.Before proceeding to this last tool, it will be useful to develop some terminology which will simplify the statement.First, the proof of Theorem 14 will proceed by induction on m = Ω(n).So when proving Theorem 14, we will be assuming that the conclusion of Theorem 14 holds for all integers n/p, where p divides n is prime.In particular, with m, n 1 , . . ., n r and n satisfying the hypothesis of Theorem 14 and G 1 , G 2 transitive abelian or nilpotent groups of degree n, then whenever K 1 , K 2 are transitive nilpotent or abelian groups of degree n/p -and to simplify our notation, there is no harm in assuming that p|n 1 -such that K 1 , K 2 are (m−1)-step imprimitive, then there exists Hi , where Hi is a solvable πi group of degree ni .Here ni = n i if i = 1 while n1 = n 1 /p, and πi is the set of prime divisors of ni .In this situation, we will say that n satisfies the main induction hypothesis.Lemma 13.Let n 1 , . . ., n r be positive integers such that if i = j, then gcd p|n (and without loss of generality, p|n 1 ), and π = ∪ j∈I πj for some admits a complete block system B of p blocks of size n/p, and By hypothesis, there exists , where each HB,j is a transitive solvable πj -group of degree nj .Similarly, if B = B ∈ B, then there exists , where each HB ,j is a transitive solvable πj -group of degree nj .Furthermore, we have that Π r j=1 HB,j for every B ∈ B, where each HB,j is a transitive solvable πj -group of degree nj .Note that δ −1 fix G 2 (B)δ = fix δ −1 G 2 δ (B) as δ/B = 1.For ease of notation, we will replace δ −1 G 2 δ by G 2 and thus assume without loss of generality that fix Thus G i admits complete block systems formed by the orbits of the S π-subgroup of fix G i (B), i = 1, 2. As each C B is formed by the orbits of the S π-subgroup of fix G i (B) restricted to the block B ∈ B, the orbits of the S π-subgroup of fix G i (B) form the complete block system C, i = 1, 2. Hence C is a block system of G i , i = 1, 2, and so by Lemma 4, C is a complete block system of G 1 , G 2 .
We now prove the main result of this section.Theorem 14.Let n 1 , . . ., n r be positive integers such that gcd(n i , n j • ϕ(n j )) = 1 if i = j, and π i be the set of distinct prime numbers dividing If either 1. each n i is a prime-power, and G 1 , G 2 are transitive nilpotent groups of degree n such that G 1 , G 2 is m-step imprimitive, or 2. G 1 , G 2 are transitive abelian groups of degree where each H i is a transitive solvable π i -group of degree n i .
Proof.Throughout the proof, if case (1) holds, we let p i be prime such that n i = p a i i , a i 1.First suppose that case (1) holds and r = 1.Then group and so nilpotent.
In both cases, we proceed by induction on m.Suppose that m = 1.Then the only case that occurs is case (1) and r = 1.The result then follows by arguments above.Assume that the result is true for all G 1 and G 2 that satisfy the hypothesis with Ω(n) = m−1 1, and let G 1 , G 2 S n satisfy the hypothesis where Ω(n) = m.In case (1), by arguments above, we may assume that r 2. In case (2), if r = 1 then the result follows from Lemma 10, so in any case we may assume without loss of generality that r 2. Let B 1 be a complete block system of G 1 , G 2 consisting of p i blocks of size n/p i , where p i |m i .Then B 1 is a complete block system of both G 1 and G 2 .As both G 1 and G 2 are nilpotent, G 1 /B 1 and G 2 /B 1 are nilpotent.We conclude that both G 1 /B 1 and G 2 /B 1 are p i -groups.Hence there exists δ 1 ∈ G 1 , G 2 such that G 1 , δ −1 G 1 δ /B 1 has order p i .We thus assume without loss of generality that G 1 , G 2 /B 1 has order p i .
. By Lemma 13 and the induction hypothesis, there exists 2 G 2 δ 2 admits a complete block system C of n i blocks of size n/n i formed by the S π -subgroups of G 1 and G 2 .We thus assume without loss of generality that G 1 , G 2 admits C as a complete block system.Similarly, by Lemma 13 and the induction hypothesis, there exists 3 G 2 δ 3 admits a complete block system D formed by the orbits of an S π i -subgroup of fix G k (B 1 ), k = 1, 2, (we remark that if n i is prime, then D is trivial).We thus also assume without loss of generality that G 1 , G 2 admits D as well. If where Ω(n i ) = a i , and Hence by the induction hypothesis there exists , where H j S n j is a transitive solvable π j -group, and P i is a p i -group of degree p i .Then G 1 , δ −1 4 G 2 δ 4 /D admits a complete block system E of n/(n i /p i ) blocks of size p i , so that G 1 , δ −1 4 G 2 δ 4 admits a complete block system E of n/n i blocks of size n i . If B 1 and B 2 consists of p i p j blocks of size n/(p i p j ) for some p j |n j with j = i.Then G 1 /B 2 and G 2 /B 2 are nilpotent and transitive.We conclude that G 1 /B 2 and G 2 /B 2 are cyclic.By Theorem 1, there exists We thus assume without loss of generality that G 1 , G 2 /B 2 is cyclic.Thus G 1 , G 2 /B 2 admits a complete block system of p j blocks of size p i , so that G 1 , G 2 admits a complete block system B 1 of p j blocks of size n/p j , and by Lemma 5 fix G 1 ,G 2 (B 1 )| B is (m − 1)-step imprimitive for every B ∈ B 1 .Hence by Lemma 13, there exists δ 4 ∈ G 1 , G 2 such that G 1 , δ −1 4 G 2 δ 4 admits a complete block system E of n/n i blocks of size n i formed by the orbits of an S π i -subgroup of fix G k (B 1 ), k = 1, 2. Hence regardless of the value of n i , we may assume without loss of generality that G 1 , G 2 admits C and E as complete block systems.As G 1 , G 2 admits a complete block system C of n i blocks of size n/n i , G 1 , G 2 S n i S n/n i .As G 1 , G 2 also admits E as a complete block system and gcd We now consider ( 1) and (2) separately.
(1) By the induction hypothesis, we may, after a suitable conjugation, assume that The result then follows by inductively applying a Sylow Theorem and then Lemma 12.
(2) By Lemma 6 every complete block system is a normal complete block system.As L G 1 ,G 2 is a modular lattice by Lemma 11, it follows by the Jordan-Dedekind Chain Condition [6, pg. 119] that all finite chains between two elements have the same length.As G 1 , G 2 admits E as a complete block system, any maximal chain between the complete the electronic journal of combinatorics 21(3) (2014), #P3.8 block systems consisting of singletons and the complete block system consisting of one block that contain E must have length m as G 1 , G 2 is normally m-step imprimitive.We conclude that G 1 , G 2 /E is (m − a i )-step imprimitive, so by the induction hypothesis we may assume after a suitable conjugation that G 1 , G 2 /E Π r j=1,j =i H j , where H j S n j is a transitive solvable π j -group.Similarly, we may assume that G 1 , G 2 /C H i , where H i is a transitive solvable π i -group.As C ∩ E is a singleton, for every C ∈ C, E ∈ E, we have that G 1 , G 2 Π r j=1 H j .By Lemma 12, we may assume that G 1 , G 2 = Π r j=1 H j as required.The result then follows by induction.
It may be worthwhile restating Theorem 14 (1) in the following form: , the prime-power decomposition of n, be such that

Solving Sets
In this section, we further develop the terminology regarding solving sets as well as the characterizations of when a particular set is a solving set that will be needed for our main results.
Definition 16.Let G be a group and define Then G L is the left-regular representation of G.We define a Cayley object of G to be a combinatorial object X (e.g.digraph, graph, design, code) such that G L Aut(X), where Aut(X) is the automorphism group of X (note that this implies that the vertex set of X is in fact G).If X is a Cayley object of G in some class K of combinatorial objects with the property that whenever Y is another Cayley object of G in K, then X and Y are isomorphic if and only if they are isomorphic by a group automorphism of G, then we say that X is a CI-object of G in K.If every Cayley object of G in K is a CI-object of G in K, then we say that G is a CI-group with respect to K. If G is a CI-group with respect to every class of combinatorial objects, then G is a CI-group.Definition 17.Let G be a finite group.We say that S ⊆ S G is a solving set for a Cayley object X in a class of Cayley objects K if for every X ∈ K such that X ∼ = X , there exists s ∈ S such that s(X) = X , s(1 G ) = 1 G for every s ∈ S, and Aut(G) S. We say that S ⊆ S G is a solving set for a class K of Cayley objects of G if whenever X, X ∈ K are Cayley objects of G and X ∼ = X , then s(X) = X for some s ∈ S, and s(1 G ) = 1 G for every s ∈ S, and Aut(G) S. Finally, a set S is a solving set for G if whenever X, X are isomorphic Cayley objects of G in any class K of combinatorial objects, then s(X) = X for some s ∈ S, s(1 G ) = 1 G for all s ∈ S, and Aut(G) S.
Remark 18.Note that the definition of a solving set given above differs from those in [11] and [3], as here, to simplify both the statements of results and their proofs, we insist that every element of the solving set fixes 1 G .It is easy to see that α −1 G L α = G L for every α ∈ Aut(G), so the image of a Cayley object of G under a group automorphism of G is a Cayley object of G.That is, in order to test for isomorphism, automorphisms of G must be considered.However, it is not always the case that every automorphism of G needs to be considered when testing for isomorphism.For example, Cayley graphs of cyclic groups of order n each have an automorphism x → −x that is also a group automorphism of Z n , and so the image of a Cayley graph under this automorphism of Z n is itself.So, while our definition of a solving set is convenient for this paper, it does not always capture the idea behind a solving set (i.e. that it should be as small as possible) exactly, but will only necessarily include extra automorphism of G (which could then be excluded).Also note that in [11] and [3], solving sets were only defined for abelian groups.
Let X be a Cayley object of G in K.We define a CI-extension of G with respect to X, denoted by CI(G, X), to be a set of permutations in S G that each fix 1 G and whenever Lemma 19.Let G be a finite group, and X a Cayley object of G in some class K of combinatorial objects.Then CI(G, X) exists.
Proof.To show existence, we only need show that there is a set of permutations Note for X a Cayley object of G in K, CI(G, X) is not unique as if T is CI-extension of X with respect to G, then for α ∈ Aut(G), {αt : t ∈ T } is also a CI-extension of X with respect to G. The following result shows the importance of CI(G, X), as if CI(G, X) is known, then the isomorphism problem is solved.
Lemma 20.Let G be a finite group, and X a Cayley object of G in some class K of combinatorial objects.Then the following are equivalent: As S is a solving set for X, δ(X) = s(X) for some s ∈ S, and s = αt for some α ∈ Aut(G) and t ∈ T .Thus the electronic journal of combinatorics 21(3) (2014), #P3.8 and T is a CI(X, G).
2) implies 1).Let X and X be isomorphic Cayley objects of G in K. Then there exists δ ∈ S G such that δ(X) = X .As G L Aut(X ), δ −1 G L δ Aut(X).As T is a CI(X, G), there exists t ∈ T and v ∈ Aut(X) such that v The following result shows that if a solving set for X has been found, then some CI(G, X) has also been found.
Lemma 21.Let G be a group, X a Cayley object of G, and S a solving set for X.Define an equivalence relation ≡ on S by s 1 ≡ s 2 if and only if s 1 = αs 2 for some α ∈ Aut(G).Let T be a set consisting of one representative from each equivalence class of ≡.Then T is a CI(G, X).
Proof.It is straightforward to show that ≡ is indeed an equivalence relation.Choose a T as is given in the statement.Let X be a Cayley object of G isomorphic to X with δ : X → X an isomorphism.Then δ −1 G L δ Aut(X).Also, as S is a solving set for X, there exists s ∈ S such that s(X) = X so that v = δ −1 s ∈ Aut(X).Let t ∈ T such that t ≡ s so that αt = s for some α ∈ Aut(G).Then Let K be a class of combinatorial objects, and G a group.We define a CI-extension of G with respect to K, denoted by CI(G, K), to be a set of permutations in S G that each fix 1 G and whenever X ∈ K is a Cayley object of G and δ ∈ S G such that δ −1 G L δ Aut(X), then there exists t ∈ CI(G, K) and v ∈ Aut(X) such that v The proofs of the following results are straightforward.
Lemma 22.Let G be a finite group, and K a class of combinatorial objects.Then the following are equivalent: Lemma 23.Let G be a group, K a class of combinatorial objects, and S a solving set for G in K. Define an equivalence relation ≡ on S by s 1 ≡ s 2 if and only if s 1 = αs 2 for some α ∈ Aut(G).Let T be a set consisting of one representative from each equivalence class of ≡.Then T is a CI(G, K).
Let G be a finite group.We define a CI-extension of G, denoted by CI(G), to be a set of permutations in S G that each fix 1 G and whenever X ∈ K is a Cayley object of G in some class K of combinatorial objects, and δ ∈ S G such that δ the electronic journal of combinatorics 21(3) (2014), #P3.8 Repeated application of Lemma 19 for every combinatorial object X in every class K of combinatorial objects shows that CI(G) exists.The proofs of the following results are straightforward.
Lemma 24.Let G be a finite group.Then the following are equivalent: Lemma 25.Let G be a group, and S a solving set for X.Define an equivalence relation ≡ on S by s 1 ≡ s 2 if and only if s 1 = αs 2 for some α ∈ Aut(G).Let T be a set consisting of one representative from each equivalence class of ≡.Then T is a CI(G).

Applications
At the present time, the isomorphism problem has not been solved for any nilpotent group that is not abelian in any class of combinatorial objects, so there are not at this time any applications for Theorem 14 (1) (although as soon as the isomorphism problem has been solved for any nonabelian p-group in certain classes of combinatorial objects, such as color digraphs, that will change immediately).We do though have an application of Theorem 14 (2) which will not only provide new examples of CI-groups with respect to color digraphs, but also illustrate how Theorem 14 (2) generalizes the main result of [3].
The following result weakens the hypothesis (replacing normally s-step imprimitive with s-step imprimitive) of [3,Theorem 16] and generalizes this result from abelian to nilpotent groups.
Theorem 26.Let n 1 , . . ., n r be positive integers such that gcd(n i , n j • ϕ(n j )) = 1 if i = j, and π i be the set of distinct prime numbers dividing n i .Let n = n 1 • • • n r , Ω(n) = m, and G a nilpotent group of degree n.Let G = Π r i=1 N i where each N i is a π i -subgroup of G, and S(i) a solving set for N i .If 1. each n i is prime-power or G is abelian, and Proof.Let δ ∈ S G .By the hypothesis, we have that there exists φ where each L i is a transitive π i -group of degree n i , 1 i r.Let CI(N i ) be a CI-extension of N i as given by Lemma 25.As S(i) is a solving set for N i , by Lemma 24 there exists t , where C i is the complete block system of L formed by the orbits of Π r j=1,j =i L i .Let t = (t 1 , . . ., t r ) the electronic journal of combinatorics 21(3) (2014), #P3.8 and v = (v 1 , . . ., v r ).Note that t fixes 1 G .As L = Π r i=1 L i , we have that v ∈ L and t −1 G L t = v −1 ω −1 φ −1 δ −1 G L δφωv.By definition, Π r i=1 CI(N i ) is a CI-extension of G, and so by Lemma 24, S = {αt : t ∈ Π r i=1 CI(N i ), α ∈ Aut(G)} is a solving set for G. Finally, it is not difficult to see that Aut(G) = Π r i=1 Aut(N i ) as G is nilpotent and if i = j then gcd(n i , n j ) = 1, and so α = (α 1 , α 2 , . . ., α r ), α i ∈ Aut(N i ).Then αt = (α 1 t 1 , α 2 t 2 , . . ., α r t r ), α i ∈ Aut(N i ) and t i ∈ CI(N i ).Thus αt ∈ Π r i=1 S(i), S Π r i=1 S(i), and Π r i=1 S(i) is a solving set for G. Definition 27.Let Ω be a set.A k-ary relational structure on Ω is an ordered pair (Ω, U ), where U ⊆ Ω k = Π k i=1 Ω.A group G S Ω is called k-closed if G is the intersection of the automorphism groups of some set of k-ary relational structures.The k-closure of G, denoted G (k) , is the intersection of all k-closed subgroups of S Ω that contain G.
Note that a 2-closed group is the automorphism group of a color digraph.The following result of Kalužnin and Klin [7] will prove useful.
Lemma 28.Let G S X and H S Y .Let G × H act canonically on X × Y .Then (G × H) (k) = G (k) × H (k) for every k 2.
If, in Theorem 26, K is the class of k-ary relational structures, and the groups L/C i are as in the proof of Theorem 26, then by Lemma 28 we may assume that each L/C i is k-closed (although there is no reason to believe that each L/C i is a π i -subgroup -but this fact is not used in the proof of Theorem 26).Proceeding as in Theorem 26 and applying Lemma 22 instead of Lemma 24, we have the following result.
Corollary 29.Let n 1 , . . ., n r be positive integers such that gcd(n i , n j • ϕ(n j )) = 1 if i = j, and π i be the set of distinct prime numbers dividing n i .Let n = n 1 • • • n r , Ω(n) = m, and G a nilpotent group of degree n.Let G = Π r i=1 N i where each N i is a π i -subgroup of G, and S(i) a solving set for N i in the class of k-ary relational structures.If 1. each n i is prime or G is abelian, and 2. whenever δ ∈ S G there exists φ ∈ G L , δ −1 G L δ such that G L , φ −1 δ −1 G L φδ is m-step imprimitive, then Π r i=1 S(i) is a solving set for G in the class of k-ary relational structures.We now give the promised application of Theorem 14 (2) which gives new CI-groups with respect to Cayley color digraphs of a particular group.Using the results in [3], this result could be obtained but only in the special cases where the following additional arithmetic conditions hold: p does not divide q − 1, and each n i is prime.Before proceeding, we need a preliminary lemma.
Lemma 30.Let n be a positive integer with a prime divisor p|n such that n/p < p.If G is a regular group of order n, and φ ∈ S n , then there exists δ ∈ G, φ −1 Gφ such that G, δ −1 φ −1 Gφδ admits a normal complete block system with blocks of size p.
Proposition 1.4.4],we have that H/B is regular of degree m, so that fix H (B) = 1 and has order k.As Stab H (B) = fix H (B) for every B ∈ B and Stab H (B)| B is transitive [2, Exercise 1.5.6],we have that fix H (B)| B is transitive for every B ∈ B. As the blocks of B have size k, we conclude that the orbits of fix H (B) fix G (B) form B. Definition 7.
G 1 (B), fix G 2 (B) | B the electronic journal of combinatorics 21(3) (2014), #P3.8 Π r j=1 H B,j for each B ∈ B. By Lemma 12, we may assume without loss of generality that fixG 1 (B), fix G 2 (B) | B = Π r j=1 HB,j for each B ∈ B. As fix G 1 (B), fix G 2 (B) | B = Π r j=1 H B,j for each B ∈ B, fix G 1 (B), fix G 2 (B) | Bhas a normal subgroup L B with orbits of size Π j∈I nj , and so fix G 1 (B), fix G 2 (B) | B admits a complete block system C B formed by the orbits of L B , B ∈ B. Also note that the S πsubgroup of fix G i (B)| B must be contained in L B , as otherwise (fix G i (B)| B )/C B contains a nontrivial normal subgroup with orbits of size dividing Π j∈I nj , i = 1, 2. However, (fix G i (B)| B )/C B is a solvable π -subgroup and gcd(Π j∈I nj , Π j ∈I,j∈[r] nj ) = 1, i = 1, 2, a contradiction.Then fix G i (B)| B , i = 1, 2, admit complete block systems D B formed by the orbits of their unique S π-subgroups, respectively, and these complete block systems must be precisely C B , for B ∈ B. Let C = ∪ B∈B C B .Clearly a normal S π-subgroup of fix G i (B) has relatively prime order and index in fix G k (B), i = 1, 2. Hence by [6, Theorem 1.1.13],an S π-subgroup of fix