On inert subgroups of a group. (English) Zbl 1167.20319

Summary: A subgroup \(H\) of a group \(G\) is called inert if \(|H:H\cap H^g|\) is finite for all \(g\) in \(G\). If every subgroup of \(G\) is inert, then \(G\) is said to be inertial. After giving an account of the basic properties of inert subgroups, we study the structure of inertial soluble groups. A classification is obtained for the groups which are finitely generated or have finite Abelian total rank.


20E07 Subgroup theorems; subgroup growth
20F16 Solvable groups, supersolvable groups
20F05 Generators, relations, and presentations of groups
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