Prime divisors and the number of conjugacy classes of finite groups

Keller, Thomas Michael; Moretó, Alexander

doi:10.1017/s030500412300035x

Math. Proc. Camb. Phil. Soc.

2023

DOI: 10.1017/s030500412300035x

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Prime divisors and the number of conjugacy classes of finite groups

Thomas Michael Keller¹,

Alexander Moretó²

Abstract: We prove that there exists a universal constant D such that if p is a prime divisor of the index of the Fitting subgroup of a finite group G, then the number of conjugacy classes of G is at least $Dp/\log_2p$ . We conjecture that we can take $D=1$ and prove that for solvable groups, we can take $D=1/3$ .

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2024

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Cited by 2 publications

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A new lower bound for the number of conjugacy classes

Çınarcı,

Keller

2024

Proc. Amer. Math. Soc.

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In 2000, Héthelyi and Külshammer [Bull. London Math. Soc. 32 (2000), pp. 668–672] proposed that if G G is a finite group, p p is a prime dividing the group order, and k ( G ) k(G) is the number of conjugacy classes of G G , then k ( G ) ≥ 2 p − 1 k(G)\geq 2\sqrt {p-1} , and they proved this conjecture for solvable G G and showed that it is sharp for those primes p p for which p − 1 \sqrt {p-1} is an integer. This initiated a flurry of activity, leading to many generalizations and variations of the result; in particular, today the conjecture is known to be true for all finite groups. In this note, we put forward a natural new and stronger conjecture, which is sharp for all primes p p , and we prove it for solvable groups, and when p p is large, also for arbitrary groups.

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A new lower bound for the number of conjugacy classes

Çınarcı,

Keller

2024

Proc. Amer. Math. Soc.

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Classifying primitive solvable permutation groups of rank 5 and 6

Dey,

O’Neal,

Tran

et al. 2024

Journal of Group Theory

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Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let G 0 G_{0} be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of G 0 G_{0} in Ω, including the trivial orbit { α } \{\alpha\} . In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.

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Prime divisors and the number of conjugacy classes of finite groups

Cited by 2 publications

References 28 publications

A new lower bound for the number of conjugacy classes

A new lower bound for the number of conjugacy classes

Classifying primitive solvable permutation groups of rank 5 and 6

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