English
There is a group isomorphism between (M.reindex e).Group and M.Group, given by the natural transport of generators under e.
Русский
Существует изоморфизм групп между (M.reindex e).Group и M.Group, реализованный естественным переносом генераторов через e.
LaTeX
$$$(M.\mathrm{reindex}~e).\mathrm{Group} \cong M.\mathrm{Group}$$$
Lean4
/-- The simple reflections of `W` generate `W` as a group. -/
theorem subgroup_closure_range_simple : Subgroup.closure (range cs.simple) = ⊤ :=
by
have : cs.simple = cs.mulEquiv.symm ∘ PresentedGroup.of := rfl
rw [this, Set.range_comp, ← MulEquiv.coe_toMonoidHom, ← MonoidHom.map_closure, PresentedGroup.closure_range_of, ←
MonoidHom.range_eq_map]
exact MonoidHom.range_eq_top.2 (MulEquiv.surjective _)