Given the set {a₁, ... , a_n) and the words w₁, w₂, ••• , W_r ( on the a_i), let F* be the free (nonabelian multiplicative) group on the set {a₁, ... , a_n) and let M be the normal subgroup generated by the words w₁,w₂, ••• , w_r. Let N be the normal subgroup generated by all words of the form a_ia_jai^-1ai^-1.

(a) F* IM is the group defined by generators {a₁, ... , a_n} and relations {w₁ = w₂ = · · · = W_r = e}.

(b) F*/ N is the free abelian group on {a₁, ... , a_n).

(c) F* /(M V N) is (in multiplicative notation) the abelian group defined by generators {a₁, .•. , a_n) and relations {w₁ = w₂ = · · · = W_r = e}.

(d) There are group epimorphisms F* → F*/ N → F*/(M V N).