Theorems · Theorem · group theory

AddAction.IsPreprimitive.of_isTrivialBlock_of_notMem_fixedPoints

∀ {G : Type u_1} {X : Type u_2} [inst : AddGroup G] [inst_1 : AddAction G X] {a : X},
  a ∉ AddAction.fixedPoints G X →
    (∀ ⦃B : Set X⦄, a ∈ B → AddAction.IsBlock G B → AddAction.IsTrivialBlock B) → AddAction.IsPreprimitive G X

If the action is not trivial, then the trivial blocks condition implies preprimitivity (pretransitivity is automatic) (based condition)

Defined in

Mathlib.GroupTheory.GroupAction.Primitive

Cited by

1 results in Mathlib

Foundations

Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddGroupAddAction