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 Показать полностьюa 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