Free Groups and Free Products

Abstract

Among the basic properties of free groups established in Chapter 2 was the fact that every group is isomorphic with a quotient group of a free group, a fundamental result that demonstrates clearly the significance of free groups. Thus the quotient groups of free groups account essentially for all groups. By contrast subgroups of free groups are very restricted; in fact they too are free. This important fact, first proved in 1921 by Nielsen in the case of finitely generated free groups, is the principal result of the first section.