Theorem 15.1. Assuming that:

we have a short exact sequence of $G$ -modules

$0 \to A \overset{ϕ}{\to} B \overset{ψ}{\to} C \to 0 .$

Then it gives rise to a long exact sequence of abelian groups:

0 \to A^{G} \overset{ϕ}{\to} B^{G} \overset{ψ}{\to} C^{G} \overset{δ}{\to} H^{1} (G, A) \overset{ϕ^{*}}{\to} H^{1} (G, B) \overset{ψ^{*}}{\to} H^{1} (G, C) .