English
Compatibility of two quotient constructions yields an equality of equivalences; the two ways of composing equivalences agree on generators.
Русский
Совместимость двух построений фактор-групп даёт равенство эквивалентностей; две последовательные композиции совпадают на базисе.
LaTeX
$$MulAction.equivSubgroupOrbits eq_1$$
Lean4
theorem card_carrier {G : Type*} [Group G] [Fintype G] (g : G) [Fintype (ConjClasses.mk g).carrier]
[Fintype <| MulAction.stabilizer (ConjAct G) g] :
Fintype.card (ConjClasses.mk g).carrier = Fintype.card G / Fintype.card (MulAction.stabilizer (ConjAct G) g) := by
classical
rw [Fintype.card_congr <| ConjAct.toConjAct (G := G) |>.toEquiv]
rw [← MulAction.card_orbit_mul_card_stabilizer_eq_card_group (ConjAct G) g, Nat.mul_div_cancel]
· simp_rw [ConjAct.orbit_eq_carrier_conjClasses]
· exact Fintype.card_pos_iff.mpr inferInstance
