More on Landau's theorem and Conjugacy Classes

2024/06/17 by Burcu Çınarcı, Çınarcı, Burcu, Thomas Michael Keller +5 · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Graph theory and applications #Group Theory (math.GR) #Matrix Theory and Algorithms #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2406.11199

openalex publication_date 2024/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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) → ∞ as x → ∞ such that n(G) ≥ 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=C₁₁²\rtimes \rm SL(2,5), or there exists a factorization p-1 = ab with a and b positive integers, such that k_p(G) ≥ a and k_p'(G) ≥ b with equalities in both cases if and only if G=C_p \rtimes C_b with C_G(C_p) = C_p.

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• Multiplicities of fields of values of conjugacy classes in finite groups

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