On some questions related to integrable groups
Russell Blyth, Francesco Fumagalli, Francesco Matucci

TL;DR
This paper investigates the classification of finite almost-simple groups and 2-homogeneous subgroups of symmetric groups that are integrable, meaning they are derived subgroups of larger groups, addressing two specific open problems.
Contribution
It classifies almost-simple finite groups and 2-homogeneous subgroups of symmetric groups that are integrable, extending understanding of their structure and integrability conditions.
Findings
Almost-simple finite groups integrable within automorphism groups are classified.
All 2-homogeneous subgroups of symmetric groups that are integrable are characterized.
The classification results solve two previously posed open problems.
Abstract
A group is integrable if it is isomorphic to the derived subgroup of a group ; that is, if , and in this case is an integral of . If is a subgroup of , we say that is integrable within if for some . In this work we focus on two problems posed in [1]. We classify the almost-simple finite groups that are integrable, which we show to be equivalent to those integrable within , where is the socle of . We then classify all -homogeneous subgroups of the finite symmetric group that are integrable within .
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Taxonomy
TopicsFinite Group Theory Research · Advanced Algebra and Geometry
