Constructing group inclusions with arbitrary depth via wreath products

• Published in: Periodica mathematica Hungarica 2023-03, Vol.86 (1), p.249-265
• Main Authors: Horváth, Erzsébet, Janabi, Hayder Abbas
• Format: Article
• Language: English
• Subjects: Group theory; Inclusions; Integers; Mathematics; Mathematics and Statistics; Questions; Subgroups
• ISSN: 0031-5303; 1588-2829
• DOI: 10.1007/s10998-022-00474-6

It was proved in a paper by Burciu, Kadison and Külshammer in 2011 that the ordinary depth d ( S n , S n + 1 ) of the symmetric group S n in S n + 1 is 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 ) of groups and subgroups where the depth d ( H n , G n ) was 2 n , 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 . 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 ◃ N ◃ G such that G / N is cyclic of order ⌈ n / 2 ⌉ , N / H is cyclic of arbitrarily large prime order and d ( H , G ) = n . We investigate the series d ( H n , G n ) in the cases when the depth, d ( H 1 , G 1 ) , is 1, 2 or 3, where H n : = H 1 × G n - 1 and G n : = G 1 ≀ C n . We also prove that if H 1 = S k and G 1 = S k + 1 then d ( H n , G n ) = 2 n k - 1 .