doi10.1112S0024610706022861 CONSTRUCTIONS RELATED TO THE RéDEI PROPERTY OF GROUPS

(i) (pα, qβ) (see [4]); (ii) (p2, p) (see [7]); (iii) (p, p, p), where p 11 (see [8]);
Received 19 January 2005; 29 September 2005. 2000 Mathematics Subject Classification 20K01 (primary), 20D60 (secondary).
a small fraction of the finite abelian groups can have the R´edei property. Also it will be shown that if p, q are distinct primes then groups of type (p, p, q) have the
If G is a direct product of cyclic groups of orders t1, . . . , ts, then we will say that G is of type (t1, . . . , ts). The group G is called an elementary p-group if it is of type (p, . . . , p) for some prime p. We say that a p-group is of rank k if it is a direct product of k cyclic groups. We have to point out that a finite abelian group can be decomposed into a direct product of cyclic groups in essentially different ways. This will not cause any problem since the uniqueness of t1, . . . , ts plays no role in our arguments. There is of course a canonical type, where one insists that ti | ti+1 for each i, 1 i s − 1. However, it is more sensible to be flexible here, as one wishes to highlight the structure of each Sylow subgroup.
R´edei property.
2. A group with the R´edei property
A subset A of a finite abelian group G is called periodic if there is an element g ∈ G \ {e} such that Ag = A. Each element g with this property is called a period of A. All the periods of A together with the identity element form a subgroup H of G. In addition there is a subset C of G such that A = HC, where the product is direct. We say that a subset A of G can be replaced by A if G = A B is a factorization whenever G = AB is a factorizationact
We say that a finite abelian group does not have the R´edei property if it can be expressed as a direct product of two of its subsets such that both subsets contain the identity element and both subsets span the whole group. It will be shown that only a small fraction of the finite abelian groups can have the R´edei property. For groups of odd order an explicit list of the possible exceptions is compiled.
Initially it was not obvious if there is any group without the R´edei property. Fraser and Gordon [2] gave the first examples for p-groups without the R´edei property. Their smallest example is an elementary 5-group of rank 6. Szabo´ [6] exhibited the first cyclic group without the R´edei property. In this paper we will show that only
Assume first that |A| = p and |B| = pq. If A contains only p-elements, then A ⊂ x, y and so A = G. We may assume that A contains not only p-elements. By Proposition 3 of [5], A can be replaced by A in the factorization G = AB, where A consists of the p-components of the elements of A. From the factorization G = A B it follows that the p-components of the elements of A are distinct. This gives that the p-components of the elements of A \ {e} have order p. (The point is that none of them can be e.) There is an element a ∈ A such that |a| = pq. It follows from Satz 6 and Hilfsatz 17 of [3] that in the factorization G = AB the factor A can be replaced by
Theorem 1. Let p, q be distinct primes. Groups of type (p, p, q) have the R´edei property.
Proof. Let G be a group of type (p, p, q) and let x, y, z be basis elements of G with |x| = |y| = p and |z| = q. Let G = AB be a normalized factorization of G. We will show that A = G or B = G.
702
SA´ NDOR SZABO´
(iv) (32, 32), (3, 3, 3, 3) (see [9]); (v) (23, 23), (22, 22, 2) (see [9] and [7]); (vi) (2, . . . , 2), with rank 7 (see [1]). Here p, q are distinct primes and α, β 2 are integers.
It is a famous result of R´edei [3] that if a finite abelian group is a direct product of its normalized subsets such that each subset has a prime number of elements then at least one of the factors must be a subgroup. We will refer to this result simply as R´edei’s theorem.
J. London Math. Soc. (2) 73 (2006) 701–715 Ce 2006 London Mathematical Society
doi:10.1112/S0024610706022861
CONSTRUCTIONS RELATED TO THE RE´DEI PROPERTY OF GROUPS
If e ∈ A and e ∈ B, then we say that A and B are normalized subsets of G. If in the factorization G = AB the factors A and B are normalized subsets then we call the factorization a normalized factorization. We will use A to denote the smallest subgroup of G which contains the subset A of G. The subgroup A is also called the span of the subset A in G or the subgroup generated by A in G.