Characterization of the Group A5 × A5 × A5 by the Set of Conjugacy Class Sizes

IF 0.4 3区数学 Q4 LOGIC

Algebra and Logic Pub Date : 2025-01-22 DOI:10.1007/s10469-025-09775-4

I. B. Gorshkov, V. V. Panshin

引用次数: 0

Abstract

For a finite group G, we denote by N (G) the set of its conjugacy class sizes. Recently, the following question was posed: given any n ∈ ℕ and an arbitrary non-Abelian finite simple group S, is it true that G ≃ Sⁿ if G is a group with trivial center and N (G) = N (Sⁿ)? The answer to this question is known for all simple groups S with n = 1, and also for S ∈ {A₅, A₆}, where A_k denotes the alternating group of degree k, with n = 2. It is proved that the group A₅ × A₅ × A₅ is uniquely defined by the set N (A₅ × A₅ × A₅) in the class of finite groups with trivial center.

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共轭类大小集对群A5 × A5 × A5的表征

对于有限群G，我们用N (G)表示它的共轭类大小的集合。最近提出了一个问题：给定任意n∈n和一个任意非阿贝尔有限简单群S，如果G是一个具有平凡中心的群且n (G) = n (Sn)，那么G是否真的≃Sn ？对于所有n = 1的简单群S，以及S∈{A5， A6}，这个问题的答案都是已知的，其中Ak表示n = 2的k次交替群。证明了具有平凡中心的有限群中的群A5 × A5 × A5是由集合N （A5 × A5 × A5）唯一定义的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Algebra and Logic 数学-数学

CiteScore

1.10

自引率

20.00%

发文量

审稿时长

>12 weeks

期刊介绍： This bimonthly journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions. Algebra and Logic is a translation of ALGEBRA I LOGIKA, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences. All articles are peer-reviewed.