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 an isomorphism induced by cup product with a generator of H2(G,Z).
A Herbrand module is an A for which the cohomology groups are finite.
The quotient may be defined for a pair of endomorphisms of an Abelian group, f and g, which satisfy the condition fg = gf = 0.
Their Herbrand quotient q(f,g) is defined as if the two indices are finite.
is exact, and any two of the quotients are defined, then so is the third and[2] 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.