Commutator Subgroup - Definition

Definition

This motivates the definition of the commutator subgroup (also called the derived subgroup, and denoted G′ or G(1)) of G: it is the subgroup generated by all the commutators.

It follows from the properties of commutators that any element of is of the form

for some natural number n. Moreover, since, the commutator subgroup is normal in G. For any homomorphism f: G → H,

,

so that .

This shows that the commutator subgroup can be viewed as a functor on the category of groups, some implications of which are explored below. Moreover, taking G = H it shows that the commutator subgroup is stable under every endomorphism of G: that is, is a fully characteristic subgroup of G, a property which is considerably stronger than normality.

The commutator subgroup can also be defined as the set of elements g of the group which have an expression as a product g = g₁ g₂ ... g_k that can be rearranged to give the identity.