# There exist subgroups of any given subnormal depth for any given tuple of nontrivial groups as quotient groups in a subnormal series

## Statement

Suppose $A_1, A_2, \dots, A_n$ are nontrivial groups (some of them could be isomorphic to each other). Then, there exists a group $G$ with a subgroup $H$ such that $H$ has a subnormal series in $G$

$H = H_0 \le H_1 \le H_2 \le \dots \le H_n = G$

such that each $H_{i-1}$ is normal in $H_i$ and $H_i/H_{i-1} \cong A_i$ and such that the subnormal depth of $H$ in $G$ is exactly $n$, i.e., $H$ has no shorter subnormal series in $G$.

## Proof

For the proof, see the linked Math Overflow question page.