Topic
Capable group
About: Capable group is a research topic. Over the lifetime, 10 publications have been published within this topic receiving 84 citations.
Papers
TL;DR: In this article, the authors determine the epicenter of 2-generator groups in the nonabelian tensor square of the generator set of a 2-group and identify those with trivial epicenter.
Abstract: A group is called capable if it is a central factor group. Let ${\mathcal{P}}$ denote the class of finite $p$-groups of odd order and nilpotency class 2. In this paper we determine the capable 2-generator groups in ${\mathcal{P}}$. Using the explicit knowledge of the nonabelian tensor square of 2-generator groups in ${\mathcal {P}}$, we first determine the epicenter of these groups and then identify those with trivial epicenter, making use of the fact that a group has trivial epicenter if and only if it is capable. A capable group in ${\mathcal{P}}$ has the two generators of highest order in a minimal generating set of equal order. However, this condition is not sufficient for capability in ${\mathcal{P}}$. Furthermore, various homological functors, among them the exterior square, the symmetric square and the Schur multiplier, are determined for the 2-generator groups in ${\mathcal{P}}$.
40 citations
TL;DR: In this article, the exterior degree of a finite group G is defined as the probability for two elements g and g′ in G such that g ∧ g′ ǫ = 1.
Abstract: We introduce the exterior degree of a finite group G to be the probability for two elements g and g′ in G such that g ∧ g′ = 1, and we shall state some results concerning this concept. We show that if G is a non-abelian capable group, then its exterior degree is less than 1/p, where p is the smallest prime number dividing the order of G. Finally, we give some relations between the new concept and commutativity degree, capability, and the Schur multiplier.
24 citations
13 Jul 2005
TL;DR: In this article, it was shown that if |H' < ∞, then |H: Z(H)| < [H'| c log 2 |H| with some constant c and this bound is essentially best possible.
Abstract: A group H is called capable if it is isomorphic to G/Z(G) for some group G Let H be a capable group I M Isaacs (2001) showed that if H is finite, then the index of the centre is bounded above by some function of |H'| We show that if |H' < ∞, then |H: Z(H)| < [H'| c log 2 |H'| with some constant c and this bound is essentially best possible We complete a result of Isaacs, showing that if H' is a cyclic group, then |H: Z(H)| ≤ |H'| 2
20 citations
TL;DR: In this article, the authors characterized capable groups of order p2q, for any distinct primes p, q, where p is the central factor of some group. But they did not consider the complexity of the groups.
Abstract: A group is said to be capable if it is the central factor of some group. In this article, among other results, we have characterized capable groups of order p2q, for any distinct primes p, q. We ha...
5 citations
Journal Article•
TL;DR: In this paper, the authors classify all capable finite p-groups with derived subgroups of order p and G/G 0 of rank n 1, where G 0 is a derived subgroup of order P and G 0 of order n − 1.
Abstract: In this paper we classify all capable finite p-groups with derived subgroup of order p and G/G 0 of rank n 1. 1. Motivation Recall the famous question of P. Hall about a given group G "Can we decide that GH/Z(H) for a group H?" That is an interesting question but unfortu- nately finding necessary and sufficient conditions for a groupG to be isomorphic to H/Z(H) is not easy. If it is possible such group is called capable following to (4). It is known that which of finitely generated abelian groups are capable (see (2) for more information). Also, among the non-abelian groups, the capability of p-groups took special attention, although the structure of all non-abelian p-groups have not been characterized but the results of (1, 6) is determined the capability of two generator 2-groups. In the preset paper we are interesting to classify the capable group where G 0 is of order p and G/G 0 of rank n − 1. 2. preliminaries This section contains some definitions, theorems and lemmas which are used in main results. We assume that the notion of Schur multiplier is known, also we use the notion of epicenter and exterior center of a group without defining them. Epicenter of a group G which is denoted by Z � (G), was introduced by Beyl, Felgner, and Schmid in (3).They showed a necessary and sufficient condition for ag roup to be capable is having trivial epicenter.
3 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2020 | 1 |
| 2017 | 1 |
| 2016 | 1 |
| 2012 | 3 |
| 2010 | 2 |
| 2005 | 1 |