Perfect codes in Cayley graphs

PERFECT CODES IN CAYLEY GRAPHS

arXiv:1609.03755v1 [math.CO] 13 Sep 2016

HE HUANG, BINZHOU XIA, AND SANMING ZHOU

Abstract. We study perfect codes and related objects such as total perfect codes and perfect t-codes in Cayley graphs from the viewpoint of group rings. We give a uniform approach to such objects that enables us to obtain new results as well as generalizations of a few known results. In particular, we obtain conditions for a normal subgroup of a finite group to be a perfect code in some Cayley graph of the group. We also discuss related group-theoretic aspects of this approach and pose three open problems. Key words: perfect code; total perfect code; efficient dominating set; efficient open dominating set; Cayley graph; group ring AMS Subject Classification (2010): 05C25, 05C69, 94B99

1. Introduction Let Γ = (V (Γ ), E(Γ )) be a graph and t ≥ 1 an integer. A subset C of V (Γ ) is called a perfect t-code [19] in Γ if every vertex of Γ is at distance no more than t to exactly one vertex of C (in particular, C is an independent set of Γ ). A perfect 1-code is simply called a perfect code. A subset C of V (Γ ) is said to be a total perfect code [35] in Γ if every vertex of Γ has exactly one neighbour in C (in particular, C induces a matching in Γ and so |C| is even). In graph theory, a perfect code in a graph is also called an efficient dominating set [10] or independent perfect dominating set [20], and a total perfect code is called an efficient open dominating set [17]. The concepts above were developed from the work in [3], which in turn has a root in coding theory. In the classical setting, a q-ary code is a subset of S n , where q and n are positive integers and S n is the set of words of length n over a set S of size q. A q-ary code C ⊆ S n is called a perfect t-code [18, 33] if every word in S n is at distance no more than t to exactly one codeword of C, where the (Hamming) distance between two words is the number of positions in which they differ. Thus, the q-ary perfect t-codes of length n in the classical setting are precisely the perfect t-codes in Hamming graph H(n, q), the graph with vertex set S n and edges joining pairs of words of Hamming distance one. Beginning with [3] and [11], perfect codes in distance-transitive graphs and association schemes in general have received extensive attention in the literature. See, for example, [6, 3, 16, 28, 29]. Since Hamming graphs are distance-transitive, perfect codes in distance-transitive graphs can be thought as generalizations of perfect codes in the classical setting. Perfect codes in Cayley graphs are another generalization of perfect codes in the classical setting because H(n, q) is the Cayley graph of (Z/qZ)n with respect to {(1, 0, . . . , 0, 0), . . . , (0, 0, . . . , 0, 1)}. In general, given a group G and a subset S of G with e ∈ / S and S −1 = S, the Cayley graph Cay(G, S) of G with respect to the 1

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HUANG, XIA, AND ZHOU

connection set S is defined to be the graph with vertex set G such that x, y ∈ G are adjacent if and only if yx−1 ∈ S. From a group theoretic point of view, a perfect t-code in Cay(G, S) can be thought as a perfect packing as well as a perfect covering of G by the balls of radius t centerd at the vertices of C with respect to the graph distance. Moreover, if X is a group, G = X × · · · × X is the direct product of n copies of X, and S consists of those elements of G with exactly one non-identity coordinate, then Cay(G, S) is isomorphic to H(n, q) with q = |X|, and any subgroup of G is a group code (under the Hamming distance). So the notion of perfect codes in Cayley graphs can be also viewed as a generalization of the concept of perfect group codes [33]. Thus the study of perfect codes is of considerable interest for both group theory and coding theory. Perfect codes in Cayley graphs have received much attention in recent years. In [23] sufficient conditions for two families of Cayley graphs of quotient rings of Z[i] √ √ and Z[ρ] to contain perfect t-codes were given, where i = −1 and ρ = (1+ −3)/2, and these conditions were proved to be necessary in [36] in a more general setting. Related works can be found in [22] and [24]. In [30] it was proved that there is no perfect code in any Cayley graph of SL(2, 2f ), f > 1 with respect to a conjugationclosed connection set. In [13] necessary conditions for the existence of perfect codes in a Cayley graph with a conjugation-closed connection set were obtained by way of irreducible characters of the underlying group. In [20] connections between perfect codes that are closed under conjugation and a covering projection from the Cayley graph involved to a complete graph were given, and counterpart results for total perfect codes were obtained in [35]. Perfect codes in circulants were studied in [12, 14, 25, 26] (a circulant is a Cayley graph on a cyclic group). In [10], E-chains of Cayley graphs on symmetric groups were constructed, where an E-chain is a countable family of nested graphs each containing a perfect code. Despite the results above, perfect t-codes in Cayley graphs deserve further study due to their relevance to group theory, coding theory and graph theory, and the area is wide open. In this paper we propose an approach to perfect t-codes and total perfect codes in Cayley graphs by using group rings. The main results and the structure of the paper are as follows: • In the next section we give connections (Theorem 2.1) between perfect tcodes (and total perfect codes) and a certain kind of factorizations (called characteristic factorizations) in group rings, and prove a few basic results about such factorizations that will be used in later sections. • In section 3 we first give a necessary and sufficient condition for a normal subgroup to be a (total) perfect code in a Cayley graph (Theorem 3.2). The sufficiency of this result is proved by construction with the help of left transversals of the subgroup (Lemma 3.1). As a corollary we prove that for any normal subgroup H of a group G, if either |H| or |G/H| is odd, then H is a perfect code in a Cayley graph of G, and if |H| is even and |G/H| is odd, then H is a total perfect code in a Cayley graph of G (Corollary 3.3).


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We prove further that this sufficient condition is also necessary when G is a cyclic group (Corollary 3.4), and moreover the Cayley graph involved can be required to be connected when |G/H| ≥ 3 (Corollary 3.5), recovering a result in [31]. Another main result (Corollary 3.8) in section 3 gives a necessary and sufficient condition for a subset of (Z/pZ)n (where p is a prime) to be a perfect code in a Cayley graph of (Z/pZ)n , and a necessary and sufficient condition for a Cayley graph of (Z/pZ)n to admit a perfect code. A counterpart result for total perfect codes is also given (Corollary 3.9). • Characteristic factorizations in group rings are a generalization of group factorizations. So it is natural to investigate whether some properties of the latter can be generalized to the former in the hope of providing tools for studying perfect codes in Cayley graphs. In section 4 we focus on two properties and introduce the notion of factorization-preserving automorphisms of a group. A few basic results on such automorphisms are given in this section. • We finish the paper by posing three problems in the last section.

In the paper all groups are finite and all graphs are finite and simple. We denote by e the identity element of a group G, and set T −1 = {t−1 | t ∈ T } for any subset T of G. 2. An approach through group rings

Given a subset T of a group G, let T denote the element of the group ring Z[G] defined by X T = µT (g)g, g∈G

where µT (g) = 1 if g ∈ T and µT (g) = 0 if g ∈ G \ T . Given subsets T1 , T2 , . . . , Tk of G, as usual, define T1 T2 · · · Tk = {t1 t2 · · · tk | ti ∈ Ti , i = 1, 2, . . . , k}; if T1 = · · · = Tk = T , then we write T k in place of T1 T2 · · · Tk . The following simple results provide a useful approach to perfect t-codes and total perfect codes by way of group rings. Theorem 2.1. Let Γ = Cay(G, S) be a Cayley graph and t a positive integer. Then the following hold for any subset C of G: (a) C is a perfect t-code in Γ if and only if G = S0t · C, where S0 = S ∪ {e}; (b) C is a total perfect code in Γ if and only if G = S · C. Proof. Suppose that C ⊆ G is a perfect t-code in Γ . Then for any g ∈ G there exists exactly one h ∈ C such that gh−1 can be expressed as g1 · · · gk for some g1 , . . . , gk ∈ S0 and 1 ≤ k ≤ t. Hence G = S0t · C. Conversely, if G = S0t · C, then every element of G can be written in a unique way as g1 · · · gk h such that

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g1 , . . . , gk ∈ S0 , h ∈ C and 1 ≤ k ≤ t, and hence C is a perfect t-code in Γ . This proves (a). The proof of (b) is similar. In general, it is nontrivial to construct perfect t-codes in a given graph. We give the following construction by using Theorem 2.1. Construction 2.2. Let G be a group, H a subgroup of G, and g an element of order n in G. Suppose that G = H × hgi and n is divisible by 2mt + 1 for some positive integers m and t. Then hg 2mt+1 i is a perfect t-code in Cay(G, S), where S = {hg j | h ∈ H, −m ≤ j ≤ m} \ {e}. Proof. This follows from Theorem 2.1(a) immediately.

Definition 2.3. An element of the group ring Z[G] is said to be a characteristic element if it is of the form T for some subset T of G. An expression of an element of Z[G] into the product of two characteristic elements of Z[G] is called a characteristic factorization of the element in Z[G]. Theorem 2.1 shows that the study of perfect t-codes and total prefect codes in Cayley graphs of a group G is essentially the study of characteristic factorizations of G in Z[G]. An expression G = HK with H and K subgroups of G is called a factorization of the group G. Group factorizations have been studied extensively in the literature; see [1, 2, 5, 21]. The following lemma connects group factorizations and characteristic factorizations in group rings. It can be easily proved with the help of the second isomorphism theorem in group theory. Lemma 2.4. Let G be a group, and let H and K be subgroups of G. Then G = HK if and only if |H ∩ K| G = H · K in the group ring Z[G]. In view of Lemma 2.4, we will sometimes study more generally characteristic factorizations of λG in Z[G] for an arbitrary positive integer λ. We compile a few observations about such factorizations in the following lemma. Lemma 2.5. Let G be a group, A and B be subsets of G, and λ be a positive integer. If λG = A · B, then the following hold: (a) λ|G| = |A||B|; (b) λG = uA · Bv for any u, v ∈ G; (c) λG = α(A) · α(B) for any α ∈ Aut(G).

b be a complete set of inequivalent irreducible representations of G. For any Let G function f : G → C, the Fourier transform of f is defined to be the map X b fb : ρ 7→ f (g)ρ(g), ρ ∈ G. g∈G

Theorem 2.6. Let G be a group, A and B be subsets of G, and λ be a positive integer. Then the following statements are equivalent:


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(a) A · B = λG; P P P b (b) g∈A ρ(g) g∈B ρ(g) = λ g∈G ρ(g) for every ρ ∈ G; b (c) µ cA (ρ)µc µG (ρ) for every ρ ∈ G. B (ρ) = λc

Proof. Since (a) can be rewritten as ! ! X X X g g =λ g, g∈A

g∈B

g∈G

it is readily seen that (a) implies (b). From the definition of the Fourier transforms of µA and µB one can see that (b) implies (c). It remains to prove that (c) implies (a). Define X µA (h)µB (h−1 g). f : G → C, g 7→ h∈G

By the convolution property of Fourier transforms (see [32, Theorem 4 of Chapb b ter 16]), we have µ cA (ρ)µc B (ρ) = f (ρ) for every ρ ∈ G. Suppose that (c) holds. Then dG (ρ) for every ρ ∈ G, b and therefore f = λµG by the formula of fb(ρ) = λc µG (ρ) = λµ the inverse Fourier transform (see [32, Theorem 2 of Chapter 16]). This implies ! ! X X A·B = µA (g)g µB (g)g g∈G

g∈G

X X µA (h)µB (h−1 g) = g g∈G

=

X

h∈G

f (g)g

g∈G

= λ

X

µG (g)g

g∈G

= λG, completing the proof.

3. Constructing perfect and total perfect codes from transversals As shown in Theorem 2.1, perfect t-codes and total perfect codes in Cayley graphs can be obtained from characteristic factorizations in group rings. In this section we study characteristic factorizations of G of the form A · H, where H is a normal subgroup of G and A is derived from a left transversal of H in G. Such factorizations will then be used to construct (total) perfect codes in Cayley graphs that are normal subgroups. Special attention will be given to the case when G is cyclic or elementary abelian. The following results can be easily proved. Lemma 3.1. Let G be a group, and let A and B be subsets of G.

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(a) If A is a subgroup of G, then G = A · B holds if and only if B is a right transversal of A in G. (b) If B is a subgroup of G, then G = A · B holds if and only if A is a left transversal of B in G. The following result gives a necessary and sufficient condition for a normal subgroup of a group G to be a perfect code or total perfect code in a Cayley graph of G. Theorem 3.2. Let G be a group and H a normal subgroup of G. Then the following hold: (a) H is a perfect code in some Cayley graph of G if and only if for any g ∈ G, g 2 ∈ H implies (gh)2 = e for some h ∈ H; (b) H is a total perfect code in some Cayley graph of G if and only if |H| is even and for any g ∈ G, g 2 ∈ H implies (gh)2 = e for some h ∈ H. Proof. Necessity: Suppose that H is a perfect code or a total perfect code in a Cayley graph of G. Then by Theorem 2.1 and Lemma 3.1 there exists a left transversal L of H in G such that L−1 = L. Let g be an element of G with g 2 ∈ H. Since L is a left transveral, there exists an element g1 of L such that g ∈ g1 H. Then gH = g1 H and g −1H = Hg −1 = Hg1−1 = g1−1H as H is normal in G. However, g 2 ∈ H implies that g −1 H = gH. Hence g1−1H = g1 H, which implies g1−1 = g1 since g1−1 ∈ L in view of L−1 = L. Now g12 = e, and from g ∈ g1 H we obtain that g1 = gh for some h ∈ H. It follows that (gh)2 = e. This together with the fact that the size of any total perfect code must be even proves the “only if” part in statements (a) and (b). Sufficiency: Suppose that H is a normal subgroup of G such that for any g ∈ G, g 2 ∈ H implies (gh)2 = e for some h ∈ H. Denote n(T ) = |{t ∈ T | t−1 ∈ / T }|

for any subset T of G. In the following we prove by construction that there exists a left transversal L of H in G such that n(L) = 0. Start with any left transversal L0 of H in G. If n(L0 ) = 0, then set L = L0 and we are done. Assume n(L0 ) > 0. Then there exists g1 ∈ L0 such that g1−1 ∈ / L0 . Since L0 is a left transversal of H in G, there exists a unique element g2 of L0 such that g1−1 ∈ g2 H. We now construct a left transversal L1 of H in G such that n(L1 ) < n(L0 ). Consider the case g1 = g2 first. In this case there exists h ∈ H such that (g1 h)2 = e as g12 ∈ H, and we set L1 = (L0 \ {g1 }) ∪ {g1 h}. It is evident that L1 is a left transversal of H in G. Since L0 is a left transversal of H in G, we have g1 h ∈ / L0 as g1 ∈ L0 . It follows that n(L0 ) = 1 + |{g ∈ L0 \ {g1 } | g −1 ∈ / L0 }|

= 1 + |{g ∈ L0 \ {g1 } | g −1 ∈ / L0 \ {g1 }}| = 1 + n(L0 \ {g1 })


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and n(L1 ) = |{g ∈ L0 \ {g1 } | g −1 ∈ / L1 }|

= |{g ∈ L0 \ {g1 } | g −1 ∈ / L1 \ {g1 }}| = n(L0 \ {g1 }) < n(L0 ).

In the case where g1 6= g2 , we set L1 = (L0 \ {g2 }) ∪ {g1−1 }. Then L1 is a left transversal of H in G. Note that g1 6= g2−1 for otherwise g1−1 = g2 ∈ L0 . Since H is normal in G, one derives from g1−1 ∈ g2 H that g1 ∈ Hg2−1 = g2−1 H. Consequently, g2−1 ∈ / L0 as L0 is a left transversal of H in G and g1 ∈ L0 . Denote U = {g1 , g2 } and V = L0 \ U. Then L1 = V ∪ {g1 , g1−1}, n(L1 ) = n(V ), and n(L0 ) = 2 + |{g ∈ V | g −1 ∈ / L0 }|

= 2 + |{g ∈ V | g −1 ∈ / L0 \ U}| = 2 + n(V ) > n(L1 ).

In either case above, L1 is a left transversal of H in G with n(L1 ) < n(L0 ). If n(L1 ) = 0, then set L = L1 . Otherwise, by the same argument as above but with L0 replaced by L1 , we can construct a left transversal L2 of H in G with n(L2 ) < n(L1 ). Repeat this procedure until a left transveral L of H in G with n(L) = 0 is obtained. Set S = L \ H. If |H| is even, then H contains an involution g0 and we set R = S ∪ {g0 }. Then S ∪ {e}, as well as R when |H| is even, are left transversals of H in G. Moreover, since n(L) = 0, both S and R are inverse-closed. By Theorem 2.1 and Lemma 3.1, H is a perfect code in Cay(G, S), and H is a total perfect code in Cay(G, R) if |H| is even. This completes the proof of the “if” part in statements (a) and (b). Corollary 3.3. Let G be a group and H a normal subgroup of G. Then the following hold: (a) If either |H| or |G/H| is odd, then H is a perfect code in some Cayley graph of G; (b) if |H| is even and |G/H| is odd, then H is a total perfect code in some Cayley graph of G. Proof. Let g be an element of G such that g 2 ∈ H. First assume that |H| is odd. Then (g 2)2k+1 = e for some nonnegative integer k. Taking h = g 2k , we have h = (g 2)k ∈ H and (gh)2 = (g · g 2k )2 = (g 2)2k+1 = e. Next assume that |G/H| is odd. Then gH is the identity element of G/H if and only if (gH)2 is. Hence g 2 ∈ H implies g ∈ H. Taking h = g −1, we have h ∈ H and (gh)2 = e2 = e. We have proved that, under the assumption of statement (a) or (b), for any g ∈ G, g 2 ∈ H implies (gh)2 = e for some h ∈ H. The result then follows from Theorem 3.2.

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The remaining part of this section is devoted to cyclic groups and elementary abelian groups. We give a few results on Cayley graphs of such groups with the help of Theorem 3.2. 3.1. Cyclic groups. In the case when G is a cyclic group, the next corollary shows that the sufficient condition in Corollary 3.3 is also necessary. Corollary 3.4. Let G be a cyclic group and H a proper subgroup of G. Then the following hold: (a) H is a perfect code in some Cayley graph of G if and only if either |H| or |G/H| is odd; (b) H is a total perfect code in some Cayley graph of G if and only if |H| is even and |G/H| is odd. Proof. We only prove (a) as the proof of (b) is similar. The “if” part is stated in Corollary 3.3. To prove the “only if” part in (a), suppose to the contrary that H is a perfect code in a Cayley graph of G but both |H| and |G/H| are even. Write |H| = 2k and |G/H| = 2m for some positive integers k and m. Since G is cyclic, we may write G = hgi. Then H = hg 2m i and (g m )2 ∈ H. By Theorem 3.2, we know that (g m h)2 = e for some h ∈ H. Write h = g 2mj , where j is a nonzero integer as H 6= {e}. Since the equation (g m h)2 = e can be rewritten as g 4mj+2m = e, 4mj + 2m is divisible by |G| = 4mk, or equivalently 2j + 1 is divisible by 2k, but this cannot happen. This contradiction shows that the “only if” part in (a) is true. Part (a) of the next corollary was obtained in [31, Theorem 3.6], but here we give a different proof under the framework of the present paper. Corollary 3.5. Let G be a cyclic group of order at least 3 and H a proper subgroup of G. Then the following hold: (a) H is a perfect code in some connected Cayley graph of G if and only if |G/H| ≥ 3 and either |H| or |G/H| is odd ([31, Theorem 3.6]); (b) H is a total perfect code in some connected Cayley graph of G if and only if |H| is even and |G/H| is odd. Proof. Let G = hgi. If H is a perfect code in a connected Cayley graph Cay(G, S) of G, then Cay(G, S) has degree |S| > 2, and so |G/H| = |S| + 1 > 3 by Theorems 2.1 and 2.5. This together with Corollary 3.4 implies the “only if” part in (a). To prove the “if” part in (a), we assume that either |G/H| > 3 is odd, or |G/H| > 4 is even and |H| is odd. This is equivalent to saying that either H = hg 2m+1 i for some integer m > 1 with 2m + 1 dividing |G|, or H = hg 2m i and |H| = 2k + 1 for some integers m > 2 and k > 0 with 2m(2k + 1) = |G|. In the former case, set S = {g ±1, . . . , g ±m }; in the latter case, set S = {g m(2k+1) , g ±1 , . . . , g ±(m−1) }. In either case we have e ∈ / S, S −1 = S, hSi = G, and S ∪ {e} is a left transversal of H in G. Hence Cay(G, S) is connected, and by Theorem 2.1 and Lemma 3.1 H is a perfect code in Cay(G, S), establishing the “if” part in (a).


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The “only if” part in (b) is given in Corollary 3.4. To prove the “if” part in (b), suppose that H = hg 2m+1 i and |H| = 2k for some integers m > 1 and k > 1 with 2k(2m + 1) = |G|. Set S = {g k(2m+1) , g ±1, . . . , g ±(m−1) }. Then e ∈ / S, S −1 = S, hSi = G, and S is a left transversal of H in G. By Theorem 2.1 and Lemma 3.1, H is a total perfect code in the connected Cayley graph Cay(G, S), as required. 3.2. Elementary abelian groups. Corollary 3.6. Let G = (Z/kZ)n , where k and n are positive integers. If k is not divisible by 4, then the following hold: (a) any subgroup of G is a perfect code in some Cayley graph of G; (b) a subgroup of G is a total perfect code in some Cayley graph of G if and only if it is of even order. Proof. Let H be an arbitrary subgroup of G. In view of Theorem 3.2, we only need to prove that for any g ∈ G, g 2 ∈ H implies (gh)2 = e for some h ∈ H. Suppose g ∈ G is such that g 2 ∈ H. Write the order of g as 2ℓ (2m+1) for some nonnegative integers ℓ and m. Then g 2m+1 has order 2ℓ , and so (g 2m+1 )2 = e since k is not divisible by 4. Let h = g 2m . Then h = (g 2)m ∈ H and (gh)2 = (g · g 2m )2 = (g 2m+1 )2 = e as required. In Theorem 3.2 and its proof we saw how to construct a (total) perfect code in a Cayley graph from a subgroup and a transversal of it. We now prove that this construction is powerful enough to produce all (total) perfect codes when the group involved is elementary abelian. We first prove the following more general result. Theorem 3.7. Let G = (Z/pZ)n be an elementary abelian group of order pn , where p is a prime and n a positive integer, and let A be a subset of G. Then the following statements are equivalent: (a) |A| = pm for some 0 ≤ m ≤ n; (b) there exists a subgroup H of G such that G = A · H; (c) there exists a subset B of G such that G = A · B. Proof. We only prove that (a) implies (b) as the other implications are apparent in view of Lemma 2.5(a). We regard G as the n-dimensional vector space over the field Fp , where the elements of G are regarded as row vectors. Suppose that |A| = pm for some 0 ≤ m ≤ n and write A = {u1 , . . . , upm }. Let U be the pm × n matrix with rows u1 , . . . , upm , and r be the rank of U. (All matrices considered here are over Fp .) Clearly, pm = |A| ≤ pr . Hence m ≤ r. Take V to be a pm × m matrix with pairwise distinct rows, and take W to be a pm × (n − m) matrix with rank r − m. It follows that the block matrix (V, W ) has rank r, and so (V, W ) = UQ for some n × n matrix Q of rank n. Set Im R=Q , 0(n−m)×m

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where Im is the identity matrix of order m and 0(n−m)×m is the (n − m) × m zero matrix. Then R is an n × m matrix of rank m. Let H = {g ∈ G | gR = 0}

be the null space of R. Then H is a subgroup of G with order pn−m . Moreover, Im Im = V, = (V, W ) UR = UQ 0(n−m)×m 0(n−m)×m

which implies that uj R 6= uk R whenever j 6= k. Accordingly, uj and uk are in different cosets of H in G for j 6= k. This together with the fact |A||H| = pn = |G| implies that A is a left transversal of H. Now by Lemma 3.1, G = A · H, which leads to statement (b). Theorems 3.7 and 2.1 together imply the following two corollaries immediately.

Corollary 3.8. Let G = (Z/pZ)n be an elementary abelian group of order pn , where p is a prime and n a positive integer. Then the following hold: (a) a subset C of G is a perfect code in some Cayley graph Cay(G, S) of G if and only if |C| = pm for some 0 ≤ m ≤ n − 1; moreover, we can choose S to be H \ {e} for some subgroup H of G; (b) a Cayley graph Cay(G, S) of G admits a perfect code C if and only if |S| = pm − 1 for some 1 ≤ m ≤ n; moreover, we can choose C to be a subgroup of G. Corollary 3.9. Let G = (Z/2Z)n be an elementary abelian group of order 2n , where n is a positive integer. Then the following hold: (a) a subset C of G is a total perfect code in some Cayley graph Cay(G, S) of G if and only if |C| = 2m for some 1 ≤ m ≤ n; moreover, we can choose S to be Hg for some subgroup H of G and element g ∈ G \ H; (b) a Cayley graph Cay(G, S) of G admits a total perfect code C if and only if |S| = 2m for some 0 ≤ m ≤ n − 1; moreover, we can choose C to be a subgroup of G. Corollary 3.9(b) is a generalization of [35, Theorem 4.1], where the Cayley graph involved is required to be connected. Note that when p is an odd prime no Cayley graph of (Z/pZ)n admits any total perfect code as pn is odd. 4. Comparison with group factorizations As shown in Lemma 2.4, characteristic factorizations in group rings can be viewed as generalizations of group factorizations. Since the latter is an important topic in group theory, one may ask whether some properties of group factorizations can be generalized to characteristic factorizations in group rings. We discuss two properties in this section. A fundamental property for group factorizations is that if G = HK for subgroups H and K of G, then we also have G = KH. We observe that similar property


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does not hold for general characteristic factorizations in group rings. For example, for G = S3 , A = {e, (1, 2)} and B = {e, (2, 3), (1, 2, 3)}, one can easily check that G = A · B 6= B · A. On the other hand, there are nonabelian groups G such that for any positive integer λ and any subsets A and B of G, λG = A·B implies λG = B ·A. We have verified by using Magma that the quaternion group G = Q8 of order 8 is such an example. A group with every subgroup normal is called a Dedekind group. Obviously, all abelian groups are Dedekind groups. A nonabelian Dedekind group is called a Hamiltonian group. By a theorem of Dedekind [9] (see also [15, Theorem 12.5.4]), Hamiltonian groups are precisely those of the form Q8 × (Z/2Z)k × H, for an integer k ≥ 0 and an abelian group H of odd order. Lemma 4.1. Suppose that G is a group such that for any subsets A and B of G, G = A · B implies G = B · A. Then G is a Dedekind group. Proof. We will show that every subgroup of G is normal. Let B be an arbitrary subgroup of G. Suppose there exist g ∈ G and b ∈ B such that g −1 bg ∈ / B. Then g and bg are in distinct left cosets of B. Take a left transversal A of B containing g and bg. It follows from Lemma 3.1 that G = A · B. On the other hand, since g and bg are in the same right coset Bg of B, A is not a right transversal of B in G, and so G 6= B · A by Lemma 3.1. This contradicts the assumption for G. Consequently, g −1 bg ∈ B for any g ∈ G and b ∈ B, which means that B is normal in G. It is unknown whether the converse of Lemma 4.1 is true or not. We will say a bit more about this in Problem 5.1. Another well-known property of group factorizations is that if G = HK for subgroups H and K of G, then G = σ(H)K = Hσ(K) for any inner automorphism σ of G. To investigate similar properties of characteristic factorizations, we introduce the following definition. The reader may compare it with Lemma 2.5(c). Definition 4.2. Let G be a group and σ an automorphism of G. We say that σ is factorization-preserving if one of the following holds: (a) for any positive integer λ and subsets A and B of G, λG = A · B implies λG = σ(A) · B. (b) for any positive integer λ and subsets A and B of G, λG = A · B implies λG = A · σ(B). We remark that if one of (a) and (b) in Definition 4.2 holds then the other holds as well, as shown in the next lemma. Lemma 4.3. Conditions (a) and (b) in Definition 4.2 are equivalent. Proof. We only prove that (a) implies (b) in Definition 4.2 as the converse is similar. Let G be a group and σ an automorphism of G such that (a) is true. Suppose that λ is a positive integer and A and B are subsets of G with λG = A · B. Then λG = λG−1 = B −1 · A−1 . Since (a) holds by our assumption, we then have λG =

12


σ(B −1 ) · A−1 = σ(B)−1 · A−1 . It follows that λG = λG−1 = A · σ(B) and hence (b) is true. A power automorphism of a group G is an automorphism of G that maps every element to some power of it. Observe that power automorphisms of G are precisely the automorphisms fixing every subgroup of G. The study of inner power automorphisms dates back to Baer [4] in 1935, and a comprehensive study of general power automorphisms can be found in [8]. Theorem 4.4. For any group G, any factorization-preserving inner automorphism of G is a power automorphism of G. Proof. Suppose that σ is a factorization-preserving inner automorphism of G. Let g be the element of G that induces σ, and A an arbitrary subgroup of G. Suppose that there exists a ∈ A such that gag −1 ∈ / A. Then e and gag −1 are in distinct right cosets of A. Take B to be a right transversal of A in G containing e and gag −1. We see that σ(B) = g −1 Bg contains e and a. As a consequence, σ(B) is not a right transversal of A in G. Now by Lemma 3.1, G = A · B 6= A · σ(B), contradicting the assumption that σ is factorization-preserving. Hence gag −1 ∈ A for any a ∈ A. This implies that gAg −1 = A and so A = g −1 Ag. Therefore, σ is a power automorphism of G. Corollary 4.5. Suppose that G is a group with trivial center. Then any factorizationpreserving inner automorphism of G is the identity. Proof. Denote by Zj the jth center of G. Since the center of G is trivial by our assumption, we have Z1 = {e} and Z2 = {x ∈ G | ∀y : x−1 y −1 xy ∈ Z1 } = {x ∈ G | ∀y : x−1 y −1 xy = e} = Z1 = {e}. Let σ be a factorization-preserving inner automorphism of G induced by g ∈ G. By Theorem 4.4 we know that σ is a power automorphism. Hence g lies in the normalizer of every subgroup of G. By [27], g ∈ Z2 . Therefore, g = e and so σ is the identity. A power automorphism is said to be universal [8] if it maps every element to the same power. For any abelian group G, the map g 7→ g n is a universal power automorphism of G if n and |G| are coprime. Our next result shows that this automorphism is factorization-preserving. Theorem 4.6. Let G be an abelian group and n a positive integer coprime to |G|. Then the map σ : G → G, g 7→ g n is a factorization-preserving automorphism of G. Proof. Write G = hg1i × · · · × hgk i with hgj i = Z/mj Z for 1 ≤ j ≤ k. As n is coprime to |G| = m1 · · · mk , n is coprime to each mj for 1 ≤ j ≤ k. This implies that σ is an automorphism of G. Let λ be a positive integer and A and B subsets of G such that λG = A · B.


13

Denote ζj = exp(2πi/j) for any positive integer j. Let ρ be a nontrivial irreducible representation of G. Then there exist integers n1 , . . . , nk such that ρ:

n1 a1 nk ak · · · ζm . g1a1 · · · gkak 7→ ζm 1 k

G → C,

Let m be the least common multiple of m1 , . . . , mk . Clearly, ρ(g) ∈ Q(ζm ) for any g ∈ G. Since gcd(n, m) = 1, there exist ℓ ∈ Z and τ ∈ Gal(Q(ζm )/Q) such that ℓ nℓ ≡ 1 (mod m) and τ (ζm ) = ζm . Since ρ is nontrivial, by Theorem 2.6 we deduce from λG = A · B that ! ! X X X ρ(g) ρ(g) = λ ρ(g) = 0. g∈A

This means that either X

g∈B

P

g∈A

g∈G

ρ(g) = 0 or X

ρ(g) =

g∈A

P

g∈B

ρ(g) =

σ(g)∈σ(A)

=

X

=

g∈σ(A)

One can see that

P

g∈A



ρ(g ℓ ) =



g∈σ(A)



ρ(g)

ρ(σ −1 (g))

X

(ρ(g))ℓ

g∈σ(A)



τ (ρ(g)) = τ 

ρ(g) = 0 if and only if

X

X

g∈σ(A)

g∈σ(A)

X

ρ(g) = 0. Note that

X

ρ(g)

g∈B

!

P

X

g∈σ(A)

g∈σ(A)

=0=λ



ρ(g) .

ρ(g) = 0. Consequently, X

ρ(g).

g∈G

Next let ρ be a trivial irreducible representation of G. Since λG = A · B, we have λ|G| = |A||B|. Hence   ! X X X  ρ(g) ρ(g) = |σ(A)||B| = |A||B| = λ|G| = λ ρ(g). g∈σ(A)

g∈B

Thus far we have shown that

g∈G

P

g∈σ(A) ρ(g)

P

P ρ(g) = λ g∈G ρ(g) for every g∈B

irreducible representation ρ of G. By virtue of Theorem 2.6, we then conclude that λG = σ(A) · B. This proves that σ is factorization-preserving. An immediate consequence of Theorem 4.6 is the following result, which was essentially known to Vuza in different terms [34, Theorem 2.3]. Corollary 4.7. Every automorphism of a cyclic group is factorization-preserving.

14


5. Open problems Problem 5.1. Investigate whether the converse of Lemma 4.1 is true or not for any Dedekind group. More explicitly, investigate whether the following statements are equivalent for any group G: (a) G is a Dedekind group; (b) for any subsets A and B of G, G = A · B implies G = B · A; (c) for any positive integer λ and subsets A and B of G, λG = A · B implies λG = B · A. Problem 5.2. Characterize subgroups H of a group G such that H is a (total) perfect code in a connected Cayley graph of G. Problem 5.3. Classify (inner) factorization-preserving automorphisms. Acknowledgements. The second author was supported by Australian Research Council grant DP150101066. The third author was supported by Australian Research Council grant FT110100629 and an MRGSS grant of the University of Melbourne. References [1] B. Amberg, S. Franciosi and F. de Giovanni, Products of Groups, Clarendon Press, Oxford, 1992. [2] B. Amberg and L. Kazarin, Factorizations of groups and related topics. Sci. China Ser. A, 52 (2009), no. 2, 217–230. [3] N. L. Biggs, Perfect codes in graphs, J. Combin. Theory Ser. B, 15 (1973), 289–296. [4] R. Baer, Der Kern, eine charakteristische Untergruppe, Compositio Math. 1 (1935), 254–283. [5] A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of Finite Groups, Walter de Gruyter GmbH & Co. KG, Berlin, 2010. [6] E. Bannai, On perfect codes in the Hamming schemes H(n, q) with q arbitrary, J. Combin. Theory Ser. A, 23 (1977), no. 1, 52C-67. [7] W. Bosma, J. Cannon and C. Playoust, The magma algebra system I: The user language, J. Symbolic Comput., 24 (1997), no. 3-4, 235–265. [8] C. D. H. Cooper, Power automorphisms of a group, Math. Z., 107 (1968), no. 5, 335–356. [9] R. Dedekind, Ueber Gruppen, deren sämmtliche Theiler Normaltheiler sind, Math. Ann., 48 (1897), no. 4, 548–561. [10] I. J. Dejter and O. Serra, Efficient dominating sets in Cayley graphs, Discrete Appl. Math., 129 (2003) 319–328. [11] P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Res. Rep. Suppl., 10, 1973. [12] Y-P. Deng, Efficient dominating sets in circulant graphs with domination number prime, Inform. Process. Lett., 114 (2014), 700–702. [13] G. Etienne, Perfect codes and regular partitions in graphs and groups, European J. Combin., 8 (1987), no. 2, 139–144. [14] R. Feng, H. Huang and S. Zhou, Perfect codes in circulant graphs, submitted, 2016. [15] M. J. Hall, The theory of groups, The Macmillan Co., New York, N.Y. 1959. [16] P. Hammond and D. H. Smith, Perfect codes in the graphs Ok , J. Combin. Theory Ser. B, 19, no. 3, 239–255. [17] T. W. Haynes, S. T. Hedetniemi and P. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc., New York, 1998.


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