Summary - Structure of finite groups and arithmetical conditions on conjugacy class sizes.


This essay is developed within the context of the Finite Group Theory. It studies the relationship 
between the structure of a group and the conjugacy class sizes of its elements. Chapter one is a 
short review of the concepts and basic results on conjugacy class sizes of a group. It collects some 
of the most important results known so far, as a starting point of our investigation. The second 
chapter contains all the preliminary results which we have needed to deal with the different problems 
on conjugacy class sizes we have treated. Our findings are collected in the last three chapters of the 
essay. The third chapter is dedicated to the study of the p-structure of the group from the conjugacy 
class sizes of its p-regular elementes. We prove an extensión of Itô's Theorem for conjugacy class 
sizes of p-regular elements, which does not make use of the p-solvability condition of the group. In 
particular, we prove thd solvability of the group. In chapter four we investigate the structure of 
groups with two conjugacy class sizes of elements of prime power order. It is proved that if G is a 
p-solvable group with class sizes of p'-elements 1 and m, then m is a product of powers of two different 
primes, p and q. We show that if m is a power of p then G has abelian p-complements, and if m is a product of 
powers of p and q, then G = PQ x A, where P and Q are a Sylow p-subgroup and a Sylow q-subgroup of G 
respectively, and A is a central subgroup of G. We also show that if G is a group with two class sizes of 
prime power order elements, then is nilpotent. In the fifth and last chapter we study the structure of normal 
subgroups of a group, under given arithmetical conditions on the G-class sizes contained in such subgroups. 
It is shown that if N is a normal subgroup of G such that the G-class sizes in N are 1 and m, for some integer 
m, then N is abelian o is a direct product of a non abelian p-group by a central subgroup of G, and therefore 
is nilpotent. We also extend some properties of the p-groups with two class sizes, for normal p-subgroups 
of a group with two G-class sizes, by proving that if G is a finite group and P is a non abelian normal p-subgroup 
with two G-class sizes then P over center of P has exponent p.