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The paper deals with direct products of subsets of a finite abelian group. The results are motivated by possible extensions of Hajos’ fundamental theorem.

It was initiated by the second author to investigate in which groups the left and right stabilizers of subsets have equal order. First we prove that if the left and right stabilizers of sets of prime power size are equal order then the group is supersolvable. We also characterize those 2-groups which satisfy this property for p = 2. We show that if in a finite group, the left and right stabilizers of sets of prime power size have equal order, then the commutator subgroup is abelian. Finally we characterize hamiltonian groups with the help of one-sided stabilizers.

In earlier papers it has been shown that certain different types of conditions on the factors in a factorization of a finite abelian group by its subsets lead to the conclusion that one factor must be a subgroup. In this paper the common generalization is proved that this result still holds even if different factors satisfy different types of condition. It is also shown that one condition may be weakened without effecting the conclusion.

It is a well-known theorem of Frobenius that a finite group G has a normal p-complement if and only if two elements of its Sylow p-subgroup that are conjugate in G are already conjugate in P. This result was generalized by Brauer and Suzuki, see e.g. [2], from Sylow to Hall subgroups using additional conditions, namely if H is a Hall $\pi$ -subgroup of G and two elements of H that are conjugate in G are already conjugate in H and each elementary $\pi$ -subgroup in G can be conjugated into H, then G has a normal $\pi$ -complement. In this paper we generalize the theorem of Frobenius from Sylow to Hall subgroups under different conditions, the conjugacy condition is restricted only for elements of odd prime order and elements of order 2 and 4 in H, on the other hand we assume that H has a Sylow tower. This also generalizes a result of Zappa, see [8], saying that if H is a Hall $\pi$ -subgroup of G with a Sylow tower, and two elements of H that are conjugate in G are already conjugate in H, then G has a normal $\pi$ -complement. As a corollary we get a weakening of the conditions of another result of Zappa, saying that if a finite group has a Hall- $\pi$ -subgroup H with a Sylow tower and H possesses a set of complete right coset representatives, which is invariant under conjugation by H, then G has a normal $\pi$ -complement. In the end we generalize the theorem of Brauer and Suzuki in another direction, namely assuming that G has a solvable Hall $\pi$ -subgroup and every elementary $\pi$ -subgroup of G can be conjugated into it, and if two elements of H of prime power order in H that are conjugate in G are already conjugate in H, then G has a normal $\pi$ -complement. In this paper all groups are finite. For basic definitions the reader is referred to [6].

So far there has been elementary proof for Frobenius's theorem only in special cases: if the complement is solvable, see e.g. [3], if the complement is of even order, see e.g. [6]. In the first section we consider the case, when the order of the complement is odd. We define a graph the vertices of which are the set K# of elements of our Frobenius group with 0 fixed points. Two vertices are connected with an edge if and only if the corresponding elements commute. We prove with elementary methods that K is a normal subgroup in G if and only if there exists an element x in K# such that all elements of K# belonging to the connected component C of K# containing x are at most distance 2 from c and NG(C) is not a -group, where is the set of prime divisors of the Frobenius complement of G. In the second section we generalize the case when the order of the complement is even, proving that the Frobenius kernel is a normal subgroup, if a fixed element a of the complement, the order of which is a minimal prime diviso...

Suppose that the finite abelian group G is a direct product of the subsets A1,..., An such that for each i, there is a subgroup Hi of G with |Ai| = |Hi| ≤ |Ai ∩ Hi| + p − 2, where p is the least prime factor of |G|. A. D. Sands proved that then Ai = Hi for some i. We prove that the same conclusion holds if |Hi| = |Ai| ≤ |Ai ∩ Hi| + pi − 2 for each i, where pi is the least prime factor of |Ai|. As generalizations of earlier results we prove two similar results.

If a finite cyclic group is a direct product of its subsets such that the cardinality of one factor is a product of two primes and the others are of prime cardinalities, then at least one of the factors is a direct product of a subset and a proper subgroup of the group. This settles a 30 years old problem of A. D. Sands,.

then we say that the product A1 ... An ,is a k-factorization of G. When the product A1 "" An is direct then it is a 1-factorization of G and will be called simply a factorization of G. The subset A of G is defined to be cyclic if it is of form {e~ a, a2, . . . ,at-l}.

Redei′s theorem asserts that if a finite abelian group is a direct product of subsets of prime cardinality, then at least one of the factors is periodic. A theorem of A. D. Sands and S. Szabo states that if a finite elementary 2-group is factored into subsets of cardinality four, then at least one of the factors is periodic. As a common generalization of these results we prove that if a finite abelian group whose 2-component is elementary is factored into subsets whose cardinalities are of prime or four, then at least one of the factors must be periodic.

The paper contains two theorems generalizing the theorems of Huppert concerning the characterization of supersolvable and p -supersolvable groups, respectively. The first of these gives a new approach to prove Huppert's first named result. The second one has numerous applications in the paper. The notion of balanced pairs is introduced for non-conjugate maximal subgroups of a finite group. By means of them some new deep results are proved that ensure supersolvability of a finite group. Introduction We recall Huppert's characterizations for ( p -)supersolvable groups. (i) Let p be some prime. A finite group is p-supersolvable iff it is p-solvable and the index of any maximal subgroup is either p or coprime to p . (ii) A finite group is supersolvable iff all maximal subgroups of it have prime index . (See in [10, 9.2–9.5 Satz], pp. 717–718.) Among others it immediately follows that the class (formation) of finite supersolvable groups is saturated, i.e. the supersolvability of G /Φ( G ) is equivalent to the supersolvability of G itself. Result (ii) turned out to be of fundamental importance and it inspired a long series of further achievements. Concentrating to various characterizations of finite supersolvable groups by means of the index of maximal subgroups or the existence of cyclic supplements to maximal subgroups we mention [7], [12] and [15] from the past; cf. also [16] (or [6, Thm. 2.2], p 483).

The paper deals with decomposition of a finite abelian group into a direct product of subsets. A family of subsets, the so-called uniquely complemented subsets, is singled out. It will be shown that if a finite abelian group is a direct product of uniquely complemented subsets, then at least one of the factors must be a subgroup. This generalizes Hajos's factorization theorem.

Hajos proved that if a finite abelian group is a direct product of cyclic subsets, then at least one of the factors is a subgroup of the group. A cyclic subset consists of the “first some consecutive elements” of a cyclic subgroup. We give a generalization for Hajos' theorem. The proof of this generalization is simpler and the steps are better motivated than the proof of the original result.

Hajos theorem asserts that if a finite abelian group is expressed as a direct product of cyclic subsets of prime cardinality, then at least one of the factors must be a subgroup. (A cyclic subset is a ‘front end’ of a cyclic subgroup.) A.D. Sands proved that if a finite cyclic group is the direct product of subsets each of which has cardinality a power of a prime, then at least one of the factors is a direct product of some subset and a nontrivial subgroup. We prove that the same conclusion holds if a general finite abelian group is factored as a direct product of cyclic subsets of prime cardinalities and general subset of cardinalities that are powers of primes provided that the components of the group corresponding to these latter primes are cyclic.

Two results on factorization of finite abelian groups are proved using combinato- rial character free arguments. The first one is a weaker form of Redei's theorem and presented only to motivate the method. The second one is an extension of Redei's theorem for elemen- tary2-groups, which was originally proved by means of characters. a1;1 a1;nDa2;1 a2;n; a1;1;a2;12A1;:::;a1;n;a2;n2An imply thata1;1D a2;1;:::;a1;nD a2;n. If the productA1 An is direct and if it is equal toG, then we say thatGDA1 An is a factorization ofG. A subset A of G is called normalized if e 2 A. A factorization G D A1 An is called normalized if each Ai is a normalized subset of G. Redei (2) has proved the following result. Let G be a finite abelian group and let G D A1 An be a normalized factorization ofG. If eachjAij is a prime, then at least one of the factors A1;:::;An must be a subgroup ofG. Examples show that the condition that each factor has a prime number of elements cannot be dropped from Redei's theorem. However for elementary 2-groups Sands and Szabo (3) proved the following generalization. Let G be a finite elementary 2- group and let G D A1 An be a normalized factorization of G. If eachjAij D 4, then at least one of the factorsA1;:::;An is a subgroup ofG. In this paper we will present an elementary combinatorial argument to verify a weaker version of Redei's theorem for elementaryp-groups, wherep is an odd prime.

will be denoted briefly by [g, q] and will be called a simplex provided the positive integer q is not greater than the order of g. Here q is the length, g is the generator and qg is the terminating element of the simplex. We denote the order of g by Ig[, and the generatum of the subset A of G, that is the smallest subgroup which contains A, will be denoted by (A). 9 The most well-known result on the field of factorization of finite abelian groups i s the so-called Haj6s' theorem which asserts that in a simplex factorization of a finite abelian group one of the simplices must be a subgroup. In this.theorem the lengths of the simplices may be reduced to primes so the following result of L. Rddei can be viewed as a generalisation for it. In a normed factorization of a finite abelian grOUP by its subsets of prime cardinalities one of the factors " as a subgroup. The algebraic form of the so-called Keller's conjecture proposes a different generalisation of Haj6s' theorem. Name!y, if

In 1930 0. H. Keller [4] conjectured that if translates of a closed n-dimensional cube tile the n-space, then in this cube system there exist two cubes having a common (n - 1 )-dimensional face. In 1949 G. Hajos [3] gave the following group theoretical equivalent for this conjecture. If G is a finite additive abelian group and

In a recent paper [l] K . CORRADI and A. HAJNAL proved that if a finite graph without multiple edges contains at least 3k vertices and the valency of every vertex is at least 2k, where k is a positive integer, then the graph contains k independent circuits, i . e . the graph contains as a subgraph a set of k circuits no two of which have a vertex in common . The present paper contains extensions of this theorem. In a recent paper [2] P . ERDŐS and L. POSA proved, among other things, that if a finite graph with or without loops and multiple edges contains n vertices and at least n + 4 edges, then the graph contains two circuits without an edge in common . The present paper contains analogous results for planar graphs . We adopt the following notation : O k denotes a graph consisting of k independent circuits, kO denotes a graph consisting of k or more circuits no two of which have an edge in common . If q is a graph then 'V (q) denotes the set of vertices of q, `V i(q ) denotes the set of vertices of q having valency i in q (i being a non-negative integer), T 5 i (q), Ty i (q) denote the set of vertices of 4 having valency i and ' i, respectively, and &(q) denotes the set of edges of c~ . The valency of the vertex x in the graph will be denoted by v (x, C~) . I ? (() I will be denoted by V (q), J & (q) I by E(q) etc. In this notation the theorem Of CORRADI and HAJNAL quoted above states that if q is a finite graph without multiple edges and if V(q) 3k and T -2k_ 1 (4) =0, then q Ok ; and the theorem of ERDOS and POSA quoted above states that if is a finite graph and E(C) _V(q) + 4, then qD 2 O .