 Their Herbrand quotient "q"("f","g") is defined as

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.