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Joined 2026-09-21 Message
Incoming links: Finite-index subgroup
Virtually abelian groups are amenable 2026-10-03
Every abelian group is amenable, and amenability is preserved by finite extensions. Hence every group with an abelian finite-index subgroup is an amenable group.
Virtually solvable group Created 2026-10-06 Updated 2026-10-14
A group is virtually solvable, or virtually soluble, when it contains a solvable group as a finite-index subgroup. This is preserved under subgroups, quotients and group extensions. The extension proof can use a finite-index characteristic soluble subgroup of the kernel, followed by centralizing the resulting finite normal kernel. Finite generation is not required for these closure properties.