More on Landau's theorem and Conjugacy Classes

TL;DR

This paper establishes new bounds on the number of conjugacy classes related to prime elements in finite groups, proving the existence of a function that guarantees a minimum number of such classes based on group size.

Contribution

It introduces two novel results on conjugacy class counts in finite groups, including a function bounding the maximum number of classes of nontrivial p-elements.

Findings

Existence of a function f(x) with f(x)→∞ as x→∞ such that n(G) ≥ f(|G|) for all finite groups G.

Characterization of groups with specific conjugacy class counts related to prime divisors, including a unique structure for certain cases.

Identification of conditions under which the number of conjugacy classes of p-elements and p'-elements meet bounds, with explicit group structures.

Abstract

In this paper we present two new results on the number of certain conjugacy classes of a finite group. For a finite group $G$, let $n(G)$ be the maximum of $k_{p}(G)$ taken over all primes $p$ where $k_{p}(G)$ denotes the number of conjugacy classes of nontrivial $p$-elements in $G$. Using a recent theorem of Giudici, Morgan and Praeger, we prove that there exists a function $f(x)$ with $f(x) \to \infty$ as $x \to \infty$ such that $n(G) \geq f(|G|)$ for any finite group $G$. Let $G$ be a finite group, and let $p$ be a prime dividing $|G|$. Let $k_{p'}(G)$ denote the number of conjugacy classes of elements of $G$ whose orders are coprime to $p$. We show that either $p=11$ and $G=C_{11}^2\rtimes \text{\rm SL}(2,5)$, or there exists a factorization $p-1 = ab$ with $a$ and $b$ positive integers, such that $k_{p}(G) \geq a$ and $k_{p'}(G) \geq b$ with equalities in both cases if and only if $G=C_p \rtimes C_b$ with $C_G(C_p) = C_p$.