= Herbrand quotient of a finite module
{c}
{title2=$h_G(M)=1\quad(|M|<\infty)$}
For a finite <module> acted on by a cyclic <group>, write $D=\sigma-1$ and $N=1+\sigma+\cdots+\sigma^{n-1}$. The two <Tate cohomology> <group> sizes are $|\ker D|/|NM|$ and $|\ker N|/|DM|$. The equalities $|NM|=|M|/|\ker N|$ and $|DM|=|M|/|\ker D|$ show that these sizes are equal. The exact-sequence multiplication rule for the <Herbrand quotient> consequently makes it unchanged by finite-index modifications of lattices.
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