The composition quotient groups belonging to two composition series of a finite group G are, apart from their sequence, isomorphic in pairs. In other words, if I subset H_s subset ... subset H_2 subset H_1 subset G is one composition series and I subset K_t subset ... subset K_2 subset K_1 subset G is another, then t = s, and corresponding to any composition quotient group K_j/K_(j + 1), there is a composition quotient group H_i/H_(i + 1) such that K_j/K_(j + 1) ≅H_i/H_(i + 1). This theorem was proven in 1869-1889.

butterfly lemma | composition series | finite group | isomorphic groups