I've looked around a lot and couldn't find much help (at least that I could understand) on this question - it is 1.45 in Lang's Algebra book:
Let $G$ be a cyclic group of order $n$, generated by $\sigma$. Assume $G$ acts on an abelian group $A$ as groups s.t. $\sigma(x+y) = \sigma(x)+\sigma(y)$ for $x,y \in A$, and let $f,g: A \to A$ be the homomorphisms defined as:
$f(x) = \sigma\,x - x $ and $g(x) = x + \sigma\,x + \cdots + \sigma^{n-1}\,x$
Herbrand quotient is given as $q(A) = (A_f: A^g)/(A_g:A^f)$, provided both indices are finite. And, $A_f$ and $A^f$ are the kernel and image of map $f$. Assume $B$ is a subgroup of $A$ s.t. $GB \subset B$. Then
a.) Define in a natural way an operation of $G$ on $A/B$
b.) Prove that $q(A) = q(B)\,q(A/B)$ Hint: consider complex: $E: 0 \to A_g \to A \overset{g}{\to} A \overset{f}{\to} A \overset{g}{\to} A^g \to 0$
Hint: $K(A): \cdots A_i \overset{d_i}{\to} A_{i+1} \overset{d_{i+1}}{\to} \cdots$ where $A_i = A$ for all $i$ and $d^i = f$ if $i$ is even and $d^i = g$ if $i$ is odd. Similarily consider $K(B)$ and $K(A/B)$. Examine long exact sequence on cohomology associated to the exact sequence of complexes $0 \to K(B) \to K(A) \to K(A/B) \to 0$. Keep in mind complexes $K$ are periodic so the long exact sequence will also be periodic, of the form $H^0(K(B)) \to H^0(K(A)) \to H^0(K(A/B)) \to H^1(K(B)) \to H^1(K(A)) \to H^1(K(A/B))$
c.)If $A$ is finite, then $q(A) = 1$.
So I fiddled around with $C_n$ groups and their subgroups to see the homomorphism work, and I can think of the isomorphism theorems - making me think of the quotient group - but I am stuck on the first part of the question.
I would appreciate any help with this!!