Classification of solvable groups with a given property (original) (raw)

Classification of non-solvable groups with a given property

Proceedings - Mathematical Sciences, 2015

In this paper, we classify the finite non-solvable groups satisfying the following property P 5 : their orders of representatives are set-wise relatively prime for any 5 distinct non-central conjugacy classes.

Finite Groups with Five Non-Central Conjugacy Classes

2017

‎Let G be a finite group and Z(G) be the center of G‎. ‎For a subset A of G‎, ‎we define kG(A)‎, ‎the number of conjugacy classes of G that intersect A non-trivially‎. ‎In this paper‎, ‎we verify the structure of all finite groups G which satisfy the property kG(G-Z(G))=5, and classify them‎.

On the Solvability of Groups with Four Class Sizes

Journal of Algebra and Its Applications, 2012

It is shown that if the set of conjugacy class sizes of a finite group G is {1, m, n, mn}, where m, n are positive integers which do not divide each other, then G is up to central factors a {p, q}-group. In particular, G is solvable.

On solvable groups with one vanishing class size

Proceedings, 2020

Let G be a finite group, and let cs(G) be the set of conjugacy class sizes of G. Recalling that an element g of G is called a vanishing element if there exists an irreducible character of G taking the value 0 on g, we consider one particular subset of cs(G), namely, the set vcs(G) whose elements are the conjugacy class sizes of the vanishing elements of G. Motivated by the results in [2], we describe the class of the finite groups G such that vcs(G) consists of a single element under the assumption that G is supersolvable or G has a normal Sylow 2-subgroup (in particular, groups of odd order are covered). As a particular case, we also get a characterization of finite groups having a single vanishing conjugacy class size which is either a prime power or square-free.

A sufficient conditon for solvability of finite groups

arXiv (Cornell University), 2017

The following theorem is proved: Let G be a finite group and π e (G) be the set of element orders in G. If π e (G) ∩ {2} = ∅; or π e (G) ∩ {3, 4} = ∅; or π e (G) ∩ {3, 5} = ∅, then G is solvable. Moreover, using the intersection with π e (G) being empty set to judge G is solvable or not, only the above three cases. 1 Introduction Let G be a finite group. We have two basic sets: |G| and π e (G). There are many famous works about |G| in the history of group theory. The set π e (G)

New Trends in Characterization of Solvable Groups

webdoc.sub.gwdg.de

Abstract. We give a survey of new characterizations of finite solvable groups and the solvable radical of an arbitrary finite group which were obtained over the past decade. We also discuss generalizations of these results to some classes of infinite groups and their analogues for ...