Every finite group has a composition series

Prove that every finite group has a composition series.


Let G be a finite group.

If |G| = 1, then G \cong 1, so that G has a trivial composition series.

Suppose that every group of order less than or equal to n has a composition series, and let G be a group of order n+1. If G is simple, then 1 \vartriangleleft G is a composition series for G. If G is not simple, then G has a nontrivial normal subgroup N. Now |N| \leq n and |G/N| \leq n, so that both N and G/N have composition series; say 1 = H_1 \vartriangleleft H_2 \vartriangleleft \cdots \vartriangleleft H_{k-1} \vartriangleleft H_k = N and 1 = K_1/N \vartriangleleft K_2/N \vartriangleleft \cdots \vartriangleleft K_{\ell-1}/N \vartriangleleft K_\ell/N = G/N. Then 1 = H_1 \vartriangleleft H_2 \vartriangleleft \cdots \vartriangle H_k = K_1 \vartriangleleft \cdots \vartriangleleft K_{\ell-1} \vartriangleleft K_\ell = G is a composition series for G. By induction, every finite group has a composition series.

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