In this note, we give an upper bound for the number of elements from the interval $[1,p^{1/4\sqrt{e}+\unicode[STIX]{x1D716}}]$ necessary to generate the finite field $\mathbb{F}_{p}^{\ast }$ with $p$ an odd prime. The general result depends on the distribution of the divisors of $p-1$ and can be used to deduce results which hold for almost all primes.

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].

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.

In 1990, Howie and McFadden showed that every proper two-sided ideal of the full transformation monoid $T_n$, the set of all maps from an $n$-set to itself under composition, has a generating set, consisting of idempotents, that is no larger than any other generating set. This fact is a direct consequence of the same property holding in an associated finite $0$-simple semigroup. We show a correspondence between finite $0$-simple semigroups that have this property and bipartite graphs that satisfy a condition that is similar to, but slightly stronger than, Hall's condition. The results are applied in order to recover the above result for the full transformation monoid and to prove the analogous result for the proper two-sided ideals of the monoid of endomorphisms of a finite vector space.