Every finite group G of order greater than one possesses a finite series of subgroups, called a composition series, such that I⊲H_s ⊲...⊲H_2 ⊲H_1 ⊲G, where H_(i + 1) is a maximal subgroup of H_i and H⊲G means that H is a normal subgroup of G. A composition series is therefore a normal series without repetition whose factors are all simple. The quotient groups G/H_1, H_1/H_2, ..., H_(s - 1)/H_s, H_s are called composition quotient groups.

butterfly lemma | finite group | Jordan-Hölder theorem | normal series | normal subgroup | quotient group | subgroup