I am trying to understand Stallings' Theorem for lower central series. Here is the statement:
Say we have groups $A, B$ with lower central series $A=A_1, A_2, ...$ and $B=B_1, B_2, ...$ respectively. Let $\phi : A \rightarrow B$ be a group homomorphism inducing an isomorphism $H_1(A) \rightarrow H_1(B)$ and an epimorphism $H_2(A) \rightarrow H_2(B)$. Then for all $n=1, 2, ...$, $f$ induces an isomorphism $A/A_n \rightarrow B/B_n$.
Here's the problem: what is meant by $H_i(A)$ and $H_i(B)$? I'm familiar with homology, but my understanding is that one speaks of the homology of a chain complex. What is the homology of a group? Is there a chain complex at play here?