On the prime graph of a finite group with unique nonabelian composition factor

Результат исследования: Научные публикации в периодических изданияхстатьярецензирование

Аннотация

We say that finite groups are isospectral if they have the same sets of orders of elements. It is known that every nonsolvable finite group G isospectral to a finite simple group has a unique nonabelian composition factor, that is, the quotient of G by the solvable radical of G is an almost simple group. The main goal of this paper is to prove that this almost simple group is a cyclic extension of its socle. To this end, we consider a general situation when G is an arbitrary group with unique nonabelian composition factor, not necessarily isospectral to a simple group, and study the prime graph of G, where the prime graph of G is the graph whose vertices are the prime numbers dividing the order of G and two such numbers r and s are adjacent if and only if (Formula presented.) and G has an element of order rs. Namely, we establish some sufficient conditions for the prime graph of such a group to have a vertex adjacent to all other vertices. Besides proving the main result, this allows us to refine a recent result by Cameron and Maslova concerning finite groups almost recognizable by prime graph.

Язык оригиналаанглийский
Страницы (с-по)3447-3452
Число страниц6
ЖурналCommunications in Algebra
Том50
Номер выпуска8
Ранняя дата в режиме онлайн11 февр. 2022
DOI
СостояниеОпубликовано - 2022

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