But while the factor group of C is always abelian, C itself need not be, in which case we can look for the commutators of C and the subgroup in Cthey form; such are called higher order commutator subgroups (also derived subgroups). If the original group G is finite then this process must obviously terminate; either we find a subgroup whose commutator subgroup is itself, or we reach an abelian commutator subgroup whose own commutator subgroup is simply . For example the tetrahedral group A4 has a commutator subgroup isomorphic to the 4-group, which is abelian. Conversely the icosahedral group A5 has no (proper) normal subgroups whatsoever, and its commutator subgroup is A5.
This gets within sight of the unsolvability of the quintic, via results which this paper can only mention lightly. `Solvable' groups are ones which have a chain of subgroups, each normal in the next larger subgroup, each with an abelian factor group, with the chain terminating in I. A5 corresponds to some quintic equation, and is not solvable, having no normal subgroups to even start a chain.