Zn×n

Zn×n (usually written Z_n × Z_n) denotes the direct product of two copies of the cyclic group of order n. It is the finite abelian group consisting of all ordered pairs (a,b) with a,b in Z_n, with addition performed componentwise modulo n. The group has order n^2 and is isomorphic to Z_n ⊕ Z_n.

In concrete terms, its elements are (a,b) where a,b ∈ {0,1,...,n−1}, and the sum (a,b) + (a′,b′) is (a+a′ mod n, b+b′ mod n). The group is not cyclic for any n>1, and its exponent is n, meaning every element has order dividing n. It is a two-dimensional Z_n-module, and its automorphism group is GL_2(Z_n), the group of invertible 2×2 matrices with entries modulo n, acting by linear transformations on the pair (a,b).

Structure and decomposition: Zn×n is already in invariant-factor form as Z_n ⊕ Z_n. If n factors as n = ∏ p_i^{a_i}, then Zn×n decomposes as the direct sum over primes: ⊕_i (Z_{p_i^{a_i}} × Z_{p_i^{a_i}}). For the special case where n is prime p, Z_p × Z_p is a 2-dimensional vector space over the field F_p, and its subgroups of order p correspond to the lines through the origin; there are p+1 such subgroups.

Applications and context: Zn×n is a standard example in the study of finite abelian groups and serves as a simple model of a discrete two-dimensional torus or lattice modulo n. It also appears in coding theory and cryptography contexts as a finite module over Z_n and as a test case for automorphism groups and subgroup structure.