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feat(GroupTheory/GroupAction/FixedPoints): characterise when there are no fixed points by an action #38285

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feat(GroupTheory/GroupAction/FixedPoints): characterise when there are no fixed points by an action #38285

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8 changes: 8 additions & 0 deletions Mathlib/GroupTheory/GroupAction/FixedPoints.lean
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Expand Up @@ -119,6 +119,14 @@ theorem smul_fixedBy (g h : G) :
ext a
simp_rw [Set.mem_smul_set_iff_inv_smul_mem, mem_fixedBy, mul_smul, smul_eq_iff_eq_inv_smul h]

lemma fixedBy_eq_empty_iff [IsRightCancelMul M] {m : M} :
fixedBy M m = ∅ ↔ m ≠ 1 := by
simp [MulAction.fixedBy, Set.eq_empty_iff_forall_notMem]

lemma fixedBy_op_eq_empty_iff [IsLeftCancelMul M] {m : M} :
fixedBy M (MulOpposite.op m) = ∅ ↔ m ≠ 1 := by
simp [MulAction.fixedBy, Set.eq_empty_iff_forall_notMem]
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@tb65536 tb65536 Apr 21, 2026 •
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Should the names somehow reflect that the action is on M itself? Otherwise these names sound like they would apply to a general MulAction?


end FixedPoints

section Pointwise
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