Сибирский федеральный университет

Тип публикации: научное издание

Год издания: 2015

Идентификатор DOI: 10.1017/CBO9781316227343.027

Аннотация: Introduction Our terminology and notation are mostly standard (see, for example, [1, 2]). We use the term “group” to mean “finite group.” Let ? be a set of primes. Denote by ?_ the set of primes not in ?. Given a natural n, we denote by ?(n) the set of prime divisors of n. A natural number n with ?(n) ? ? is called a ?-number, and a group G such that ?(G) ? ? is called a ?-group. For a group G, the set ?(G) = ?(|G|) is the prime spectrum of G. A subgroup H of a group G is called a ?-Hall subgroup if ?(H) ? ? and ?(|G: H|) ? ?’. Thus, if ? consists of a single prime p then a ?-Hall subgroup is exactly a Sylow p-subgroup. A Hall subgroup is a ?-Hall subgroup for some set ? of primes. A group G is prime spectrum minimal if ?(H)? ?(G) for every proper subgroup H of G. We say that G is a group with Hall maximal subgroups if every maximal subgroup of G is a Hall subgroup. It is easy to see that every group with Hall maximal subgroups is prime spectrum minimal. A group G is a group with complemented maximal subgroups if for every maximal subgroup M of G, there exists a subgroup H such that MH = G and M ? H = 1. The study of groups with Hall maximal subgroups was started in 2006 by Levchuk and Likharev [3] and Tyutyanov [4], who established that a nonabelian simple group with complemented maximal subgroups is isomorphic to one of the groups PSL2(7) PSL3(2), PSL2(11) or PSL5(2). In all these groups, every maximal subgroup is a Hall subgroup. In 2008, Tikhonenko and Tyutyanov [5] showed that the nonabelian simple groups with Hall maximal subgroups are exhausted up to isomorphism by the groups PSL2(7), PSL2(11), and PSL5(2). © Cambridge University Press 2015.

Журнал: Groups St Andrews 2013

Номера страниц: 428-435

• Maslova N.V. (N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian, Academy of Sciences, Ekaterinburg, Russian Federation, Ural Federal University named after the first President of Russia B.N. Yeltsin, Ekaterinburg, Russian)
• Revin D.O. (S.L. Sobolev Institute of Mathematics of the Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russian Federation, Novosibirsk State University, Novosibirsk, Russian Federation)

• Scopus