Normal subgroups and p-regular G-class sizes (original) (raw)

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.

Certain relations between p-regular class sizes and the p-structure of p-solvable groups

Journal of the Australian Mathematical Society, 2004

Let G be a finite p-solvable group for a fixed prime p. We study how certain arithmetical conditions on the set of p-regular conjugacy class sizes of G influence the p-structure of G. In particular, the structure of the p-complements of G is described when this set is {1, m, n} for arbitrary coprime integers m, n > 1. The structure of G is determined when the noncentral p-regular class lengths are consecutive numbers and when all of them are prime powers.

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.

Solvability of normal subgroups and GGG-class sizes

Publicationes Mathematicae Debrecen, 2013

We study the solvability of a normal subgroup N of a finite group G having exactly three G-conjugacy class sizes. We show that if the set of G-class sizes of N is {1, m, mp a }, with p a prime not dividing m, then N is solvable. Thus, we get a partial extension for normal subgroups on N. Itô's theorem on the solvability of groups having exactly three class sizes.

FINITE GROUPS WITH TWO p-REGULAR CONJUGACY CLASS LENGTHS II

Bulletin of the Australian Mathematical Society, 2009

Let G be a finite group. We prove that if the set of p-regular conjugacy class sizes of G has exactly two elements, then G has Abelian p-complement or G=PQ×A, with P∈Sylp(G), Q∈Sylq(G) and A Abelian.

Nilpotency of normal subgroups having two GGG-class sizes

Proceedings of the American Mathematical Society, 2011

Let G be a finite group. If N is a normal subgroup which has exactly two G-conjugacy class sizes, then N is nilpotent. In particular, we show that N is abelian or is the product of a p-group P by a central subgroup of G. Furthermore, when P is not abelian, P/(Z(G) ∩ P) has exponent p.

Normal sections, class sizes and solvability of finite groups

Journal of Algebra, 2014

If G is a finite group, we show that any normal subgroup of G which has exactly three G-conjugacy class sizes is solvable. Thus, we give an extension for normal subgroups of the classical N. Itô's theorem which asserts that those finite groups having three class sizes are solvable, and particularly, a new proof of it is provided. In order to do this, we investigate the structure of a normal section N/K of G such that every element in N lying outside of K has the same G-class size.