Notes on abelian groups. II (original) (raw)

Elementary Abelian p-groups revisited

Bulletin of the Australian Mathematical Society, 1995

For each prime p, a Fraenkel-Mostowski model is constructed in which there are two elementary Abelian p-groups with the same cardinality that are not isomorphic.

On some Abelian p-groups and their capability

2015

A group is said to be capable if it is a central factor group; equivalently, if and only if a group is isomorphic to the inner automorphism group of another group. In this research, the capability of some abelian pgroups which are groups of order p4 and p5, where p is an odd prime are determined. The capability of the groups is determined by using the classifications of the groups.

On Sets of PP-Generators of Finite Groups

Bulletin of the Australian Mathematical Society, 2014

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.

Powerful p-groups. I. Finite groups

Journal of Algebra, 1987

In this paper we study a special class of finite p-groups, which we call powerful p-groups. In the second part of this paper, we apply our results to the study of p-adic analytic groups. This application is possible, because a finitely generated prop group is p-adic analytic if and only if it is "virtually pro-powerful." These applications are described in the introduction to the second part, while now we describe the present part in more detail. In the first section we define a powerful p-group, as one whose subgroup of pth powers contains the commutator subgroup. We give several results

On minimal non-p-closed groups and related properties

Publicationes Mathematicae Debrecen, 2011

Let p be a prime. A group is called p-closed if it has a normal Sylow p-subgroup and it is called p-exponent closed if the elements of order dividing p form a subgroup. A group is minimal non-p-closed if it is not p-closed but its proper subgroups and homomorphic images are. Similarly, a group is called minimal non-p-exponent closed if it is not p-exponent closed but all its proper subgroups and homomorphic images are. In this paper we characterize finite minimal non-p-closed groups and investigate the relationship between them and minimal non-p-exponent closed groups. In particular, we show that every minimal non-p-closed group is non-p-exponent closed and that minimal non-p-closed groups and simple minimal non-p-exponent closed groups have cyclic Sylow p-subgroups. Furthermore, given a prime p, we describe non-p-exponent closed groups of smallest order and we show that they coincide with non-p-closed groups of smallest order.