Higher Order Commutator Subgroups

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 $\{I\}$. 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.