  The third homology group of GL_n(R) is studied, where R is a `ring with many
units' with center Z(R). The main theorem states that if K_1(Z(R))_Q \simeq
K_1(R)_Q, (e.g. R a commutative ring or a central simple algebra), then
H_3(GL_2(R), Q) --> H_3(GL_3(R), Q) is injective. If R is commutative, Q can be
replaced by a field k such that 1/2 is in k. For an infinite field R (resp. an
infinite field R such that R*=R*^2), we get a better result that H_3(GL_2(R),
Z[1/2] --> H_3(GL_3(R), Z[1/2]) (resp. H_3(GL_2(R), Z) --> H_3(GL_3(R), Z)) is
injective. As an application we study the third homology group of SL_2(R) and
the indecomposable part of K_3(R).
