Pál Hegedűs

The constants of the Volterra derivation

The ring of constants of the Volterra derivation is found. Confirming a conjecture of Zielinski, it is always a polynomial ring.

Determination of conjugacy class sizes from products of characters

In Robinson [5] Robinson showed that the character degrees are determined by knowing, for all n, the number of ways that the identity can be expressed as a product of n commutators. Earlier, in Strunkov [6], Strunkov showed that the existence of characters of p-defect 0 can be determined by counting solutions to certain equations involving commutators and conjugates. In this paper, we prove analogs to Robinson’s and Strunkov’s theorems by switching conjugacy classes and characters. We show that counting the multiplicity of the trivial character in certain products of characters determines the conjugacy class sizes and existence of conjugacy classes with p-defect 0.

Groups where each element is conjugate to its certain power

This paper deals with a rationality condition for groups. Let n be a fixed positive integer. Suppose every element g of the finite solvable group is conjugate to its nth power gn. Let p be a prime divisor of the order of the group. We conclude that the multiplicative order of n modulo p is small, or p is small.