On Sets of Pp-Generators of Finite Groups

Abstract

The classes of finite groups with minimal sets of generators of fixed cardinalities, named B-groups, and groups with the basis property, in which every subgroup is a B-group, contain only p-groups and some {p, q}-groups. Moreover, abelian B-groups are exactly p-groups. If only generators of prime power orders are considered, then an analogue of property B is denoted by B pp and an analogue of the basis property is called the pp-basis property. These classes are larger and contain all nilpotent groups and some cyclic q-extensions of p-groups. In this paper we characterise all finite groups with the pp-basis property as products of p-groups and precisely described {p, q}-groups.2010 Mathematics subject classification: primary 20F05; secondary 20D10, 20D60.

‘On Sets of Pp-Generators of Finite Groups’

“…However, the proof was given only in the Frattini-free case. Thus, in [3], we proved in fact only the result below. Theorem 1.…”

“…In [3,Theorem 4.5] we gave a description of pp-matroid groups. However, the proof was given only in the Frattini-free case.…”

Sets of prime power order generators of finite groups

“…Solvable groups with the pp-embedding property were studied in [10,11]. In [7] all pp-matroid groups were de-scribed. Moreover in [7] it was proved that pp-matroid groups have the pp-basis exchange property.…”

Finite groups with the pp-embedding property

“…The class of pp-matroid groups and groups with the pp-basis property, i.e. groups whose all subgroups have property B, have been completely described (see [9]).…”

“…From definitions, we know that every group with property B has property B pp , but not conversely. For example from [7,8] we know that all nilpotent groups have property B pp (are pp-matroid). On the other hand from [10] we know that Finite groups with the pp-embedding property 3 among nilpotent groups only p-groups have property B (are matroid).…”