ON THE QUOTIENT OF A FREE GROUP BY THE COMMUTATOR OF 2 NORMAL-SUBGROUPS

[摘要] Let F be a noncyclic free group and let K and L be normal subgroups of F. In trying to describe the group F/[K, L] in terms of F/K and F/L, it is important to understand the group A = K and L/[K, L]. The main theorem gives an exact sequence into which the group A fits. The sequence reduces in special cases to Hopf's Formula for H2(F/K, Z) and to the Magnus embedding or relation sequence embedding the relation module in a free module. The sequence is 0 --> H2(S) + H2(T) --> A --> P --> DELTA(C) X(ZC) ZB --> 0, where B = F/KL, C = F/K and L, DELTA(C) denotes the augmentation ideal of C, P is a free ZB-module on a basis in one to one correspondence with a basis of F, and the sequence is of ZB-modules. As one application of this, we show that if each of K and L is the normal closure of a single element, then A is free abelian. A second application gives that if l > 2 and F(n),F(n) are the nth terms of the lower central and derived series respectively of F, then F/[F(c+1), F(l)] contains an infinite elementary abelian 2-group.

[发布日期] 1991-09-30