Note di Matematica 23

It is proved that if a finite abelian group is factored into a direct product of lacunary cyclic subsets, then at least one of the factors must be periodic. This result generalizes Hajós's factorization theorem.

Proceedings of the Edinburgh Mathematical Society, 1973

Let G be a nite abelian group of order p 4 , where p is a prime.

Proceedings of the American Mathematical Society, 1969

To appear in Advanced Research in Pure Mathematics

Acta Mathematica Academiae Scientiarum Hungaricae

w 5. p.basic subgroups of arbitrary abelian groups KULIKOV [8] introduced the notion of basic subgroups of abelian p-groups which has proved to be one of the most important notions in the theory of p-groups of arbitrary power. Basic subgroups can be defined in any module over the ring of p-adic integers, or, more generally, over any discrete valuation ring. Here we want to give a generalization of basic subgroups to any group so that it coincides with the old concept whenever the group is primary. In the general case, to every prime p, one can define p-basic subgroups where in the definition the prime p plays a distinguished role. The p-basic subgroups are not isomorphic for different primes, but are uniquely determined (up to isomorphism) by the group and the prime p. We shall see that p-basic subgroups are useful in certain investigations. Let G be an arbitrary (abelian) group l and p an arbitrary, but fixed prime. We call a subset [x~]~ea of G, not containing 0, p-independent, if for any finite subset xl .... ,x~ a relation nlxl-[-... q-nkx1~ EprG

The aim of this paper is to develope a new method to prove some classic theorems of abelian groups. In particular, we study the lattices of finite abelian groups.

Journal of Group Theory, 2000

It is proved that every group of the form G D AB with subgroups A and B, each of which is either abelian or generalized dihedral, is soluble.

In this paper, we determine the number of subgroups of group ⊗ which may be cyclic or non-cyclic by using simple number-theoretic formulae.

Journal of Group Theory

It is proved that every group of the form G = AB with two subgroups A and B each of which is either abelian or has a quasicyclic subgroup of index 2 is soluble of derived length at most 3. In particular, if A is abelian and B is a locally quaternion group, this gives a positive answer to Question 18.95 of "Kourovka notebook" posed by A.I.Sozutov.