A Note on “Conjugacy Classes Outside a Normal Subgroup

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.

Bulletin of the London Mathematical Society, 1974

Bulletin of the Iranian Mathematical Society, 2022

Let $G$ be a finite p-group. Assume that $ν(G)$ and $ν_c(G)$ denote the number of conjugacy classes of non-normal subgroups and non-normal cyclic subgroups of $G$, respectively. In this paper, we completely classify the finite p-groups with $ν_c = p$ or $p + 1$ for an odd prime number $p$. Also, we classify the groups $G$ with $ν(G) = νc(G) = p^i ,i ≥ 1$.

Siberian Mathematical Journal, 1997

Journal of Group Theory, 2009

Journal of Algebra, 1982

International Journal of Group Theory, 2017

Some of the results of this paper are part of the third author's Ph.D. thesis at the University Jaume I of Castellon, who is financially supported by a predoctoral grant of this university. The first and second authors are supported by the Valencian Government, Proyecto PROMETEOII/2015/011. The first and the third authors are also partially supported by Universitat Jaume I, grant P11B2015-77.

Journal of Algebra, 1984

ON FINITE FACTORIZABLE GROUPS 523 (I) A, with r > 5 a prime and A N A,-, . (II) M,, and either A is solvable or A N M,,. (III) M,, and either B is Frobenius of order 11 . 23 or B is cyclic of order 23 and A N M,, .

Bulletin of the Malaysian Mathematical Sciences Society, 2016

Let G be a finite Frobenius group with at most two conjugacy classes of each size. In this paper, we shall prove that G is isomorphic to S

2000

We define a Con-Cos group G to be one having a proper normal subgroup N whose cosets other than N itself are conjugacy classes. It follows easily that N = G0, the derived group of G. Most of the paper is devoted to trying to classify finite Con-Cos groups satisfy- ing the additional requirement that N has just two conjugacy

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.

Http Dx Doi Org 10 1080 00927879908826617, 2007

Journal of Group Theory, 2022

Let 𝐾 and 𝐷 be conjugacy classes of a finite group 𝐺, and suppose that we have K n = D ∪ D - 1 K^{n}=D\cup D^{-1} for some integer n ≥ 2 n\geq 2 . Under these assumptions, it was conjectured that ⟨ K ⟩ \langle K\rangle must be a (normal) solvable subgroup of 𝐺. Recently R. D. Camina has demonstrated that the conjecture is valid for any n ≥ 4 n\geq 4 , and this is done by applying combinatorial results, the main of which concerns subsets with small doubling in a finite group. In this note, we solve the case n = 3 n=3 by appealing to other combinatorial results, such as an estimate of the cardinality of the product of two normal sets in a finite group as well as to some recent techniques and theorems.

2016

Let G be a finite group. We say that the derived covering number of G is finite if and only if there exists a positive integer n such that C n = G ′ for all non-central conjugacy classes C of G. In this paper we characterize solvable groups G in which the derived covering number is finite.