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ON SETS OF PP-GENERATORS OF FINITE GROUPS

Published online by Cambridge University Press:  14 October 2014

JAN KREMPA
Affiliation:
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland email jkrempa@mimuw.edu.pl
AGNIESZKA STOCKA*
Affiliation:
Institute of Mathematics, University of Białystok, Akademicka 2, 15-267 Białystok, Poland email stocka@math.uwb.edu.pl
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Abstract

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The classes of finite groups with minimal sets of generators of fixed cardinalities, named ${\mathcal{B}}$-groups, and groups with the basis property, in which every subgroup is a ${\mathcal{B}}$-group, contain only $p$-groups and some $\{p,q\}$-groups. Moreover, abelian ${\mathcal{B}}$-groups are exactly $p$-groups. If only generators of prime power orders are considered, then an analogue of property ${\mathcal{B}}$ is denoted by ${\mathcal{B}}_{pp}$ 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.

Type
Research Article
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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