English
For a finite group G, the set of powers of x equals the set of zpowers of x.
Русский
Для конечной группы G множество степеней x совпадает с множеством z-пower x.
LaTeX
$$(powers x : Set G) = zpowers x$$
Lean4
@[to_additive]
theorem zpowers_eq_zpowers_iff {x y : G} (hx : ¬IsOfFinOrder x) : zpowers x = zpowers y ↔ x = y ∨ x⁻¹ = y :=
by
refine ⟨fun h ↦ ?_, by rintro (rfl | rfl) <;> simp⟩
have hx_mem : x ∈ zpowers y := by simp [← h]
have hy_mem : y ∈ zpowers x := by simp [h]
obtain ⟨k, rfl⟩ := mem_zpowers_iff.mp hy_mem
obtain ⟨l, hl⟩ := mem_zpowers_iff.mp hx_mem
rw [← zpow_mul] at hl
nth_rewrite 2 [← zpow_one x] at hl
have h1 := (injective_zpow_iff_not_isOfFinOrder.mpr hx) hl
rcases (Int.mul_eq_one_iff_eq_one_or_neg_one).mp h1 with (h | h) <;> simp [h.1]
