Finite composition length implies every subnormal series can be refined to a composition series

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Statement

If G is a Group of finite composition length (?), and we have a subnormal series of G of the form:

\{ e \} = H_0 < H_1 < H_2 < \dots < H_r = G

where each H_i is a proper normal subgroup of H_{i+1}, then there exists a composition series:

\{ e \} = N_0 < N_1 < \dots < N_l = G

where each H_i occurs as N_j for some j.

(Note that if the original subnormal series has repetitions, we can throw out the repetitions, and even if it does not start at the trivial group, we can insert the trivial group. Thus, there is no loss of generality in assuming that the subnormal series is of the above form).

Facts used

  1. Finite composition length is normal subgroup-closed
  2. Finite composition length is quotient-closed
  3. Lattice isomorphism theorem
  4. Zassenhaus isomorphism theorem

Proof

Proof idea

The idea is to find a composition series of H_i/H_{i-1} for each i and use the lattice isomorphism theorem to fill in subgroups between H_{i-1} and H_i that form part of a composition series for G. Thus, the main question is: does H_i/H_{i-1} have a composition series?

Hands-off proof

Given: Group G of finite composition length with a strictly ascending subnormal series H_i.

To prove: H_i can be refined to a composition series.

Proof:

  1. By facts (1) and (2), each quotient H_i/H_{i-1} has finite composition length. The lattice isomorphism theorem (Fact (3)) allows us to therefore get a series of subgroups from H_{i-1} to H_i with the factor groups being simple.
  2. Putting together all these series provides a composition series for G that refines the H_is.

Hands-on proof

Given: Group G of finite composition length with a strictly ascending subnormal series H_i.

To prove: H_i can be refined to a composition series.

Proof: The idea is to use the fact that G has some composition series of finite length to construct a composition series of finite length that refines the H_is. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]