If G is a finite cyclic group acting on a G-module A, then the cohomology groups Hn(G,A) have period 2 for n≥1; in other words
A Herbrand module is an A for which the cohomology groups are finite. In this case, the Herbrand quotient h(G,A) is defined to be the quotient
of the order of the even and odd cohomology groups.
if the two indices are finite. If G is a cyclic group with generator γ acting on an Abelian group A, then we recover the previous definition by taking f = 1 - γ and g = 1 + γ + γ2 + ... .
is exact, and any two of the quotients are defined, then so is the third and
These properties mean that the Herbrand quotient is usually relatively easy to calculate, and is often much easier to calculate than the orders of either of the individual cohomology groups.