Every normal subgroup appears in a composition series

If G is a finite group and N \leq G is normal, prove that there is a composition series of G, one of whose terms is N.


Let G be a finite group and N \leq G normal. Let 1 = H_1 \vartriangleleft \cdots \vartriangleleft H_k = N and 1/N = K_1/N \vartriangleleft \cdots \vartriangleleft K_\ell/N = G/N be composition series for N and G/N, respectively; these exist by the previous exercise. Then 1 = H_1 \vartriangleleft \cdots \vartriangleleft H_k = N = K_1 \vartriangleleft \cdots \vartriangleleft K_\ell = G is a composition series for G, one of whose terms is N.

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