invariantfactor

Invariant factor refers to a canonical set of numbers used to describe the structure of certain modules, especially finitely generated modules over a principal ideal domain (PID). For a PID R, every finitely generated R-module M can be written as a direct sum of a free part and a torsion part: M ≅ R^r ⊕ ⊕_{i=1}^t R/(d_i), where r is a nonnegative integer and the d_i are nonzero elements of R satisfying d1 divides d2, divides ..., divides dt. The sequence (d1, ..., dt) is the invariant factor decomposition of M, and the elements d_i are called the invariant factors.

In the context of finitely generated abelian groups (i.e., modules over the integers Z), the invariant factor decomposition yields a decomposition of the form G ≅ Z/d1Z ⊕ Z/d2Z ⊕ ... ⊕ Z/dtZ with d1 | d2 | ... | dt. This is unique up to isomorphism and provides a concise description of the group's structure, especially when combined with the elementary divisor decomposition, which expresses each d_i as a product of prime powers.

The invariant factors are closely related to the Smith normal form of a presentation matrix. Given a finitely presented module with a presentation matrix, transforming it to Smith normal form yields a diagonal matrix diag(d1, ..., dt) with d1 | d2 | ... | dt, revealing the invariant factors directly. The elementary divisors are the prime-power factors of the invariant factors and correspond to a secondary refinement of the same structure.

Examples illustrate the concept: a finite abelian group of order 72 with invariant factors 6 and 12 is isomorphic to Z/6Z ⊕ Z/12Z (since 6 | 12). If the factorization of the invariant factors into coprime components is applied, one may also express the group via primary (p-power) components, but the invariant factors themselves provide a single, ordered chain of divisors describing the whole torsion structure.