two_sylow_eq_kleinFour_of_card_eq_four

English

For card α = 4, any Sylow 2-subgroup of Alt α is Klein four.

Русский

Для |α| = 4 любая подгруппа Sylow 2 Alt α равна Klein Four.

LaTeX

$$$S = kleinFour(α)$ for |α| = 4$$

Lean4

theorem two_sylow_eq_kleinFour_of_card_eq_four (hα4 : Nat.card α = 4) (S : Sylow 2 (alternatingGroup α)) :
    S = kleinFour α :=
  by
  rw [kleinFour, ← Subgroup.closure_insert_one, ← Set.singleton_union, ← coe_two_sylow_of_card_eq_four hα4]
  exact S.closure_eq.symm

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