instIsSolvableOfFinite — Mathlib

English

For a finite Z-group G, the exponent of G equals the cardinality of G.

Русский

Для конечной Z-группы G экспонента равна порядку группы.

LaTeX

$$\( \operatorname{exponent}(G) = |G| \quad \text{for finite } G \text{ with } G \text{ Z-group}.$$

Lean4

instance [Finite G] [IsZGroup G] : IsSolvable G :=
  by
  rw [isSolvable_iff_commutator_lt]
  intro H h
  rw [← H.nontrivial_iff_ne_bot] at h
  rw [← H.range_subtype, MonoidHom.range_eq_map, ← Subgroup.map_commutator, Subgroup.map_subtype_lt_map_subtype]
  exact commutator_lt H

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