Constructing group inclusions with arbitrary depth via wreath products

Abstract

AbstractIt was proved in a paper by Burciu, Kadison and Külshammer in 2011 that the ordinary depth $$d(S_n,S_{n+1})$$ d ( S n , S n + 1 ) of the symmetric group $$S_n$$ S n in $$S_{n+1}$$ S n + 1 is $$2n-1$$ 2 n - 1 , so arbitrarily large odd numbers can occur as subgroup depth. Lars Kadison in 2011 posed the question if subgroups of even ordinary depth bigger than 6 can occur. Recently in a paper with Breuer we constructed a series $$(G_n,H_n)$$ ( G n , H n ) of groups and subgroups where the depth $$d(H_n,G_n)$$ d ( H n , G n ) was 2n, thus answering the question of Kadison. Here we generalize the method of that proof. The main result of this paper is that for every positive integer n there are infinitely many pairs (G, H) of finite groups such that $$d(H,G)=n$$ d ( H , G ) = n . As a corollary of its proof we get that for every positive integer n there are infinitely many triples (H, N, G) of finite solvable groups $$H\triangleleft N\triangleleft G$$ H ◃ N ◃ G such that G/N is cyclic of order $$\lceil {n/2} \rceil $$ ⌈ n / 2 ⌉ , N/H is cyclic of arbitrarily large prime order and $$d(H,G)=n$$ d ( H , G ) = n . We investigate the series $$d(H_n,G_n)$$ d ( H n , G n ) in the cases when the depth, $$d(H_1,G_1)$$ d ( H 1 , G 1 ) , is 1, 2 or 3, where $$H_n:=H_1\times G^{n-1}$$ H n : = H 1 × G n - 1 and $$G_n:=G_1\wr C_n$$ G n : = G 1 ≀ C n . We also prove that if $$H_1=S_k$$ H 1 = S k and $$G_1=S_{k+1}$$ G 1 = S k + 1 then $$d(H_n,G_n)=2nk-1$$ d ( H n , G n ) = 2 n k - 1 .