Nilpotency of normal subgroups having two 𝐺-class sizes (original) (raw)

Nilpotency of p-complements and p-regular conjugacy class sizes

Journal of Algebra, 2007

Let G be a finite p-solvable group. We prove that if the set of conjugacy class sizes of all p-elements of G is {1, m, p a , mp a }, where m is a positive integer not divisible by p, then the p-complements of G are nilpotent and m is a prime power. This result partially extends a theorem for ordinary classes which asserts that if the set of conjugacy class sizes of a finite group G is exactly {1, m, n, mn} and (m, n) = 1, then G is nilpotent.

Finite p-Groups of Nilpotency Class 3 with Two Conjugacy Class Sizes

Israel Journal of Mathematics

It is proved that, for a prime p > 2 and an integer n ≥ 1, finite p-groups of nilpotency class 3 and having only two conjugacy class sizes 1 and p n exist if and only if n is even; moreover, for a given even positive integer, such a group is unique up to isoclinism (in the sense of Philip Hall).

Normal subgroups and p-regular G-class sizes

Journal of Algebra, 2011

Let G be a finite p-solvable group and N be a normal subgroup of G. Suppose that the p-regular elements of N have exactly two G-conjugacy class sizes. In this paper it is shown that, if H is a p-complement of N, then either H is abelian or H is a product of a q-group for some prime q = p and a central subgroup of G.

An Upper Bound to the Number of Conjugacy Classes of Non-Abelian Nilpotent Groups

Journal of Mathematics and Statistics

The number of conjugacy classes of symmetric group, dihedral group and some nilpotent groups is obtained. Until now, it has not been obtained for all nilpotent groups. Although there are some lower bounds to this value, there is no non-trivial upper bound. This paper aims to investigate an upper bound to this number for all finite nilpotent groups. Moreover, the exact number of conjugacy classes is found for a certain case of non-abelian nilpotent groups.

On nilpotent subgroups containing non-trivial normal subgroups

Journal of Group Theory, 2010

Let G be a non-trivial finite group and let A be a nilpotent subgroup of G. We prove that if jG : Aj c expðAÞ, the exponent of A, then A contains a non-trivial normal subgroup of G. This extends an earlier result of Isaacs, who proved this in the case where A is abelian. We also show that if the above inequality is replaced by jG : Aj < ExpðGÞ, where ExpðGÞ denotes the order of a cyclic subgroup of G with maximal order, then A contains a non-trivial characteristic subgroup of G. We will use these results to derive some facts about transitive permutation groups.

On finite products of nilpotent groups

Archiv der Mathematik, 1994

i. Introduetion. A well-known theorem of Kegel [7] and Wielandt [9] states the solubility of every finite group G = AB which is the product of two nilpotent subgroups A and B; see [1], Theorem 2.4.3. In order to determine the structure of these groups it is of interest to know which subgroups of G are conjugate (or at least isomorphic) to a subgroup that inherits the factorization. A subgroup S of the factorized group G = AB is called prefactorized if S = (A c~ S) (B ~ S), it is called factorized if, in addition, S contains the intersection A c~ B. Generally, even characteristic subgroups of G are not prefactorized, as can be seen e.g. from Examples 1 and 2 below.

The influence of p-regular class sizes on normal subgroups

Journal of Group Theory, 2013

Let G be a finite group and N be a normal subgroup of G and suppose that the p-regular elements of N have exactly two G-conjugacy class sizes. It is shown that N is solvable and that if H is a p-complement of N , then either H is abelian or H is the product of an r-group for some prime r 6 D p and a central subgroup of G.