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Let M be a family of maximal subgroups of a group G: We say that M is irredundant if its intersection is not equal to the intersection of any proper subfamily of M: The maximal dimension of G is the maximal size of an irredundant family of maximal subgroups of G. In this paper we study a class of solvable groups, called M-groups, in which the maximal dimension has properties analogous to that of the dimension of a vector space such as the span property, the extension property and the basis exchange property. [ABSTRACT FROM AUTHOR]

A subset X of a finite group G is called g-independent if there is no proper subset Y of X such that = (X, Φ(G)>. The group G has the embedding property if every g-independent subset of G can be embedded in a minimal generating set of G. If X is a set of prime power order elements, then we say that G has the pp-embedding property. In this note we classify all finite solvable groups with the pp-embedding property. Moreover we prove that this class is equal to the class of finite solvable groups with the embedding property. [ABSTRACT FROM AUTHOR]

We consider here some notions and results about sets of generators of finite algebras motivated by the case of finite groups. We illustrate these notions by simple examples closed to lattices and semigroups. Next, we examine these notions in the case of groups with operators. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM PUBLISHER]

Abstract: In several papers finite groups with fixed cardinalities of sets of generators are investigated and sometimes are named -groups. Groups with all subgroups being -groups are called groups with the basis property. In the first part of this paper we correct and generalize some characterizations of groups with the basis property. In the second part we consider groups satisfying analogous conditions, but only for sets of generators of prime power orders. The class of groups introduced here is much larger then the class of groups with the basis property. For example, it contains all nilpotent groups. [Copyright &y& Elsevier]

This note completes the proof of the structure theorem for pp-matroid groups which was stated in our earlier paper J. Krempa and A. Stocka [‘On sets of pp-generators of finite groups’, Bull. Aust. Math. Soc.91 (2015), 241–249]. [ABSTRACT FROM PUBLISHER]

Abstract: In Section 5 of our paper there are some errors related to degrees of generalized pp-extensions. We provide here a correct version of definition and results adapted to this definition. [Copyright &y& Elsevier]