María José Felipe

Some class size conditions implying solvability of finite groups

Abstract Let G be a finite group. We show that when the conjugacy class sizes of G are {1, m, n, mn}, with m and n positive integers such that (m, n) = 1, then G is solvable. As a consequence, we obtain that G is nilpotent and that m = pa and n = qb for two primes p and q.

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

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 f1; m; ng 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 normal subgroups having two G-class sizes

First published in Proceedings of the American Mathematical Society in volume 139, number 8, August 2011, published by the American Mathematical Society

ON THE DIAMETER OF A p-REGULAR CONJUGACY CLASS GRAPH OF FINITE GROUPS

ABSTRACT Let be a finite -solvable group. Attach to the following graph : its vertices are the non-central conjugacy classes of -regular elements of , and two vertices are connected by an edge if their cardinalities are not coprime. We prove that the number of connected components of is at most 2. When is connected, then the diameter of the graph is at most 3, and when is disconnected, then each of the two components is a complete graph.

Finite Groups with Four Conjugacy Class Sizes

We determine the structure of all finite groups with four class sizes when two of them are coprime numbers larger than 1. We prove that such groups are solvable and that the set of class sizes is exactly {1, m, n, mk}, where m, n > 1 are coprime numbers and k > 1 is a divisor of n.

The structure of finite groups with three class sizes

Abstract Let G be a finite group and suppose that the set of conjugacy class sizes of G is {1, m, mn}, where m, n > 1 are coprime. We prove that m = p for some prime p dividing n – 1. We also show that G has an abelian normal p-complement and that if P is a Sylow p-subgroup of G, then |P′| = p and |P/Z(G) p | = p 2. We obtain other properties and determine completely the structure of G.

Normal subgroups and class sizes of elements of prime power order

If G is a finite group and N is a normal subgroup of G with two Gconjugacy class sizes of elements of prime power order, then we show that N is nilpotent.

ON THE SOLVABILITY OF GROUPS WITH FOUR CLASS SIZES

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.

Prime Factors of π-Partial Character Degrees and Conjugacy Class Sizes of π-Elements

Let G be a finite solvable group. We prove that any prime dividing any irreducible π-partial character degree of G divides the size of some conjugacy class of π-elements of G. Under certain hypothesis, we show that if two distinct primes r and s both divide some irreducible π-partial character degree, then there exists a conjugacy class of π-elements whose size is divisible by rs.

Square-free class sizes in products of groups

Abstract We obtain some structural properties of a factorised group G = A B , given that the conjugacy class sizes of certain elements in A ∪ B are not divisible by p 2 , for some prime p. The case when G = A B is a mutually permutable product is especially considered.