Stallings-Stammbach sequence

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The Stallings-Stammbach sequence is a slight generalization of the Stallings exact sequence. Suppose we have a short exact sequence of groups:

1 \to N \to G \to Q \to 1

Suppose that B is a Q-module, i.e., an abelian group equipped with a homomorphism of groups:

Q \to \operatorname{Aut}(B)

The Stallings-Stammbach sequence is the following short exact sequence of groups:

H_2(G;B) \stackrel{\alpha_B}{\to} H_2(Q;B) \stackrel{\beta_B}{\to} N^{\operatorname{ab}} \otimes_Q B\stackrel{\sigma_B}{\to} H_1(G;B) \stackrel{\tau_B}{\to} H_1(Q;B)

In the case that B = \mathbb{Z} with trivial action of Q on it, the Stallings-Stammback sequence becomes the Stallings exact sequence.