Splitting Groups with Basis Property

Abstract

A finite group G is called splitting or splittable if it is a union of some collections of its proper subgroups intersecting pairwise at the identity. A special kind of splitting is known to be normal splitting. Also, a group G is said to have the basis property if, for each subgroup H≤G, H has a basis (minimal generating set), and any two bases have the same cardinality. In this work, I discuss a relation between classes of finite groups that possess both normal splitting and the basis property. This paper shows mainly that any non-p-group with basis property is normal splitting. However, the converse is not true in general. A counterexample is given. It is well known that any p-group has basis property. I demonstrate some types of p-groups which are splitting as well.

Finite groups with the pp-embedding property

“…Proof. (1) Suppose that N is a normal subgroup of G such that N F (G) and F (G) N. By assumption the Frattini subgroup of F (G) is also trivial, so F (G) is a direct product of elementary abelian groups. In particular, G is solvable.…”

“…• has the embedding property if every g-independent set of G can be embedded in a g-base of G ( [1,13]);…”

On Sets of Pp-Generators of Finite Groups

“…Recall, as in [11], that G is a matroid group if G has property B and every gindependent subset of G is contained in a g-base of G. Some characterisations of matroid groups can be found in [1,2,11].…”