 Herbrand quotient

When the two indicated groups are finite, their size ratio is multiplicative on short exact sequences. It is one for finite modules and therefore unchanged by finite-index modifications. A trivial integral module $\mathbb{Z}$ has quotient $|G|$, while an induced regular lattice ${\mathbb{Z}}_p[G]$ has both Tate groups zero and quotient one. These facts permit norm-index calculations without explicitly finding every norm.

The Herbrand quotient is a concept from model theory and mathematical logic, particularly within the context of the study of formal systems and the properties of logical formulas. It generally pertains to measuring certain aspects of structures in formal theories, especially in relation to the notion of definability and algebraic properties of models. Specifically, the Herbrand quotient is defined in the context of Herbrand's theorem, which relates to the concept of Herbrand universes and Herbrand bases.