Bounding the number of classes of a finite group in terms of a prime

Abstract

Héthelyi and Külshammer showed that the number of conjugacy classes k(G) of any solvable finite group G whose order is divisible by the square of a prime p is at least (49p + 1)/60. Here an asymptotic generalization of this result is established. It is proved that there exists a constant c > 0 such that for any finite group G whose order is divisible by the square of a prime p we have k(G) ≥ cp.

The average element order and the number of conjugacy classes of finite groups

“…This paper has generated a lot of research since 2000. It was mentioned in [7] that if this bound holds for p-solvable [10] and the case p = 2 was proved in [21], but the conjecture is (as of now) open for odd p. See [24] for a detailed exposition of the results and techniques that have been used.…”

“…See the introduction of [21] for a summary of the developments in this area. In that paper, A. Maróti and I. Simion proved that there exists a constant c > 0 such that k(G) cp for any finite group G of order divisible by p 2 .…”

Prime divisors and the number of conjugacy classes of finite groups

“…The following result, which is a combination of the main results of [10,23], will allow us to consider just groups with Sylow p-subgroups of order p. THEOREM 2•2. There exists a constant c such that for any group G whose order is divisible by the square of a prime p, we have k(G) ≥ cp.…”

The average element order and the number of conjugacy classes of finite groups

“…As usual, given a finite group G we write k(G) to denote the number of conjugacy classes of G. Also, if L K are normal subgroups of G with K/L abelian, we say that K/L is an abelian section of G. Using Lemma 2.6 and Corollary 2.10 of [11], it is easy to see that if Question 1.1 had an affirmative answer, then the following question would also have an affirmative answer. See the introduction of [21] for a summary of the developments in this area. In that paper, A. Maróti and I. Simion proved that there exists a constant c > 0 such that k(G) cp for any finite group G of order divisible by p 2 .…”