English
The nth iterated wreath product G is defined by recursion: IteratedWreathProduct G 0 = PUnit and IteratedWreathProduct G (n+1) = (IteratedWreathProduct G n) ≀ᵣ G.
Русский
N-й итерированный вентовый произведение G задается рекурсивно: IteratedWreathProduct G 0 = PUnit и IteratedWreathProduct G (n+1) = (IteratedWreathProduct G n) ≀ᵣ G.
LaTeX
$$$ IteratedWreathProduct(G,n) = \begin{cases} PUnit, & n=0 \\ IteratedWreathProduct(G,n) \wr_{\mathrm{r}} G, & n>0 \end{cases} $$$
Lean4
@[to_additive]
theorem card_dvd_exponent_pow_rank' {n : ℕ} (hG : ∀ g : G, g ^ n = 1) : Nat.card G ∣ n ^ Group.rank G :=
(card_dvd_exponent_pow_rank G).trans (pow_dvd_pow_of_dvd (Monoid.exponent_dvd_of_forall_pow_eq_one hG) (Group.rank G))
