On finite groups subspectral to finite almost simple groups
A. Kh. Zhurtova,
D. V. Lytkinab,
V. D. Mazurovc a Kabardino-Balkarian State University, 173 Chernyshevsky St., Nalchik 360004, Russia
b Novosibirsk State University, 2 Pirogova St., Novosibirsk 630090, Russia
c Sobolev Institute of Mathematics SB RAS, 4 Ak. Koptyg Ave., Novosibirsk 630090, Russia
Abstract:
Spectrum
$\omega(G)$ of a finite group
$G$ is the set of element orders of
$G$. This set is closed under divisibility of its elements, therefore it can be uniquely defined by its subset
$\mu(G)$ consisting of maximal under divisibility elements of
$\omega(G)$. Two groups are said to be isospectral, if their spectra coincide. A finite group
$G$ is called recognizable by spectrum in the class of finite groups (or recognaizable), if every finite group whose spectrum coincides with
$\omega(G)$ is isomorphic to
$G$. In a recent survey dedicated to recognizability of finite groups, an unsolved question is mentioned about recognizability of symmetric group
$S_{10}$ of all permutations of degree
$10$. Difficulty of this problem is, in particular, due to a wast number of finite simple groups, which are subspectral to
$S_{10}$, i. e. simple groups whose spectra are subsets of
$\omega(S_{10})$. This paper gives a method of finding all groups subspectral to a given group, and for every alternating group
$L$ the list of subspectral to
$S_{10}$ covers of
$L$ are given, whose basements are irreducible modules of representations of
$L$ over finite fields.
Key words:
spectrum, recognizability by spectrum, cover.
UDC:
512.542
MSC: 20D05 Received: 12.04.2025
DOI:
10.46698/w4978-1776-4637-t
Fulltext:
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