On the capability of nonabelian groups of order p4 (original) (raw)
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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.
A note on p-groups of order ≤ p 4
Proceedings - Mathematical Sciences, 2009
In [1], we defined c(G), q(G) and p(G). In this paper we will show that if G is a p-group, where p is an odd prime and |G| ≤ p 4 , then c(G) = q(G) = p(G). However, the question of whether or not there is a p-group G with strict inequality c(G) = q(G) < p(G) is still open.
On capable groups of order p2q
Communications in Algebra, 2020
A group is said to be capable if it is the central factor of some group. In this paper, among other results we have characterized capable groups of order p 2 q, for any distinct primes p, q, which extends Theorem 1.2 of S. Rashid, N. H. Sarmin, A. Erfanian, and N. M. Mohd Ali, On the non abelian tensor square and capability of groups of order p 2 q, Arch. Math., 97 (2011), 299-306. We have also computed the number of distinct element centralizers of a group (finite or infinite) with central factor of order p 3 , which extends Proposition 2.
2020
Following P. Hall a soluble group whose Sylow subgroups are all abelian is called A-group. The purpose of this article is to give a new and shorter proof for a criterion on the capability of A-groups of order p^2q, where p and q are distinct primes. Subsequently we give a sufficient condition for n-capability of groups having the property that their center and derived subgroups have trivial intersection, like the groups with trivial Frattini subgroup and A-groups. An interesting necessary and sufficient condition for capability of the A-groups of square free order will be also given.
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
Capability of a pair of groups
The Bulletin of the Malaysian Mathematical Society Series 2
A group G is called capable if it is the group of inner automorphisms of some group E. Capable pairs are defined in terms of a relative central extension. In this paper we introduce the precise center for a pair of groups and prove that this subgroup makes a criterion for characterizing the capability of the pair. We also show that our result sharpens the obtained result in this area. A complete classification of finitely generated abelian capable pairs will also be given.
A Simple Classification of Finite Groups of Order p2q2
2018
Suppose G is a group of order p^2q^2 where p>q are prime numbers and suppose P and Q are Sylow p-subgroups and Sylow q-subgroups of G, respectively. In this paper, we show that up to isomorphism, there are four groups of order p^2q^2 when Q and P are cyclic, three groups when Q is a cyclic and P is an elementary ablian group, p^2+3p/2+7 groups when Q is an elementary ablian group and P is a cyclic group and finally, p + 5 groups when both Q and P are elementary abelian groups.