Hall classes in linear groups

A well-known theorem of Philip Hall states that if a group 𝐺 has a nilpotent normal subgroup 𝑁 such that is nilpotent, then 𝐺 itself is nilpotent. We say that a group class 𝔛 is a if it contains every group 𝐺 admitting a nilpotent normal subgroup 𝑁 such that belongs to 𝔛. Examples have been given in...

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Published inJournal of group theory Vol. 27; no. 2; pp. 383 - 412
Main Authors de Giovanni, Francesco, Trombetti, Marco, Wehrfritz, Bertram A. F.
Format Journal Article
LanguageEnglish
Published De Gruyter 01.03.2024
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Summary:A well-known theorem of Philip Hall states that if a group 𝐺 has a nilpotent normal subgroup 𝑁 such that is nilpotent, then 𝐺 itself is nilpotent. We say that a group class 𝔛 is a if it contains every group 𝐺 admitting a nilpotent normal subgroup 𝑁 such that belongs to 𝔛. Examples have been given in [F. de Giovanni, M. Trombetti and B. A. F. Wehfritz, Hall classes of groups, to appear] to show that finite-by-𝔛 groups do not form a Hall class for many natural choices of the Hall class 𝔛. Although these examples are often linear, our aim here is to prove that the situation is much better within certain natural subclasses of the universe of linear groups.
ISSN:1433-5883
1435-4446
DOI:10.1515/jgth-2023-0063