Pronormality of Hall Subgroups in Their Normal Closure

E. P. Vdovin, M. N. Nesterov, D. O. Revin

Результат исследования: Научные публикации в периодических изданияхстатья

2 Цитирования (Scopus)

Аннотация

It is known that for any set π of prime numbers, the following assertions are equivalent: (1) in any finite group, π-Hall subgroups are conjugate; (2) in any finite group, π-Hall subgroups are pronormal. It is proved that (1) and (2) are equivalent also to the following: (3) in any finite group, π-Hall subgroups are pronormal in their normal closure. Previously [10, Quest. 18.32], the question was posed whether it is true that in a finite group, π-Hall subgroups are always pronormal in their normal closure. Recently, M. N. Nesterov [7] proved that assertion (3) and assertions (1) and (2) are equivalent for any finite set π. The fact that there exist examples of finite sets π and finite groups G such that G contains more than one conjugacy class of π-Hall subgroups gives a negative answer to the question mentioned. Our main result shows that the requirement of finiteness for π is unessential for (1), (2), and (3) to be equivalent.

Язык оригиналаанглийский
Страницы (с-по)451-457
Число страниц7
ЖурналAlgebra and Logic
Том56
Номер выпуска6
DOI
СостояниеОпубликовано - 1 янв 2018

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