isOfFinOrder_iff_zpow_eq_one — Mathlib

English

The minOrder is bounded by the natCard of any nontrivial finite subgroup: minOrder ≤ card(s).

Русский

minOrder ограничено кардиналом любого ненулевого конечного подгруппы: minOrder ≤ card(s).

LaTeX

$$$s \neq \{0\} \Rightarrow \minOrder(G) \le \mathrm{Nat.card}(s)$$$

Lean4

@[to_additive]
theorem isOfFinOrder_iff_zpow_eq_one {G} [DivisionMonoid G] {x : G} : IsOfFinOrder x ↔ ∃ (n : ℤ), n ≠ 0 ∧ x ^ n = 1 :=
  by
  rw [isOfFinOrder_iff_pow_eq_one]
  refine
    ⟨fun ⟨n, hn, hn'⟩ ↦ ⟨n, Int.natCast_ne_zero_iff_pos.mpr hn, zpow_natCast x n ▸ hn'⟩, fun ⟨n, hn, hn'⟩ ↦
      ⟨n.natAbs, Int.natAbs_pos.mpr hn, ?_⟩⟩
  rcases (Int.natAbs_eq_iff (a := n)).mp rfl with h | h
  · rwa [h, zpow_natCast] at hn'
  · rwa [h, zpow_neg, inv_eq_one, zpow_natCast] at hn'

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