Theorems · Theorem · group theory

MulAction.IwasawaStructure.commutator_le

∀ {M : Type u_1} [inst : Group M] {α : Type u_2} [inst_1 : MulAction M α] (IwaS : MulAction.IwasawaStructure M α)
  [MulAction.IsQuasiPreprimitive M α] (N : Subgroup M) [nN : N.Normal],
  MulAction.fixedPoints (↥N) α ≠ Set.univ → commutator M ≤ N

The Iwasawa criterion : If a quasiprimitive action of a group G on X has an Iwasawa structure, then any normal subgroup that acts nontrivially contains the group of commutators.

Defined in

Mathlib.GroupTheory.GroupAction.Iwasawa

Cited by

4 results in Mathlib

Foundations

Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

GroupMulActionMulAction.IsQuasiPreprimitiveSubgroup.Normal