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

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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.

Related facts

Proof

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

References