Groups whose set of vanishing elements is exactly a conjugacy class (original) (raw)

Groups with Restricted Conjugacy Classes

2002

Let F C0 be the class of all finite groups, and for each non- negative integer n define by induction the group class F Cn+1 consisting of all groups G such that for every element x the factor group G/CG(h xi G) has the property F Cn. Thus F C1-groups are precisely groups with finite conjugacy classes, and the class F Cn obviously contains all finite groups and all nilpotent groups with class at most n. In this paper the known theory of F C-groups is taken as a model, and it is shown that many properties of F C-groups have an analogue in the class of F Cn-groups.

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.

On Finite Number of Conjugacy Classes in Groups

Communications in Algebra, 2014

The work is inspired by an article of M. Herzog, P. Longobardi, and M. Maj, who considered groups with a nite number of innite conjugacy classes. Their main results were obtained under assumption that the F C-center is of nite index in the group. We consider here innite groups with a nite number of conjugacy classes of any size (F N CC-groups). Hence the F C-center in our case will be nite, but of innite index in the group. Among results on these groups we give a criterion for a wreath product of F N CC-groups to be an F N CC-group.

On Finite Groups Whose Every Normal Subgroup is a lJnion of the Same Number of Conjugacy Classes

2002

Abstract. Let G be a finite group and ,A/C denote the set of non-trivial proper normal subgroups of G. An element K of NC is said to be n-decomposable if K is a union of n distinct conjugacy classes of G. In this paper, we investigate the structure of finite groups G in which G' is a union of three distinct conjugacy classes of G. We prove, under certain conditions, G is a Flobenius group with kernel G/ and its complement is abelian.

Monomiality of finite groups with some conditions on conjugacy classes

Journal of Mathematical Sciences, 2009

We present some arithmetical-type conditions on the set of conjugacy classes of a finite group that are sufficient for the monomiality of the group, i.e., for the property that all its irreducible complex characters are induced by linear characters of subgroups.

Groups whose Non-Normal Subgroups Have Finite Conjugacy Classes

Pre-publicaciones del …, 2002

The structure of groups for which the set of non-normal subgroups has prescribed properties has been investigated by several authors. The first step was of course the description of groups in which all subgroups are normal; it is well known that such groups either are abelian or ...