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 *C*they 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 *A*_{4} has a
commutator subgroup isomorphic to the 4-group, which is abelian. Conversely
the icosahedral group *A*_{5} has no (proper) normal subgroups whatsoever, and
its commutator subgroup is *A*_{5}.

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*. *A*_{5} corresponds
to some quintic equation, and is not solvable, having no normal subgroups to
even start a chain.