conjMap — Mathlib

English

If hs is preprimitive, then by fixing subgroups on intersections with g·s we retain pre primitiveness.

Русский

Если hs пре-примитивна, то пересечение с g·s сохраняет пре-примитивность.

LaTeX

$$$\text{isPreprimitive\_ofFixingSubgroup\_inter}$$$

Lean4

/-- Conjugation induces an equivariant map between the SubMulAction of
the stabilizer of a point and that of its translate. -/
@[to_additive existing]
def conjMap {g : G} {a b : α} (hg : b = g • a) :
    MulActionHom (stabilizerEquivStabilizer hg) (ofStabilizer G a) (ofStabilizer G b)
    where
  toFun x := ⟨g • x.val, fun hy ↦ x.prop (by simpa [hg] using hy)⟩
  map_smul' := fun ⟨k, hk⟩ ↦ by simp [← SetLike.coe_eq_coe, subgroup_smul_def, stabilizerEquivStabilizer, ← smul_assoc]

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