Zqm×n

Zqm×n denotes the direct product of two cyclic structures of orders qm and n. In many algebra texts Z_m denotes the cyclic group of order m (often identified with the ring Z/mZ, the integers modulo m). Therefore Z_qm × Z_n consists of ordered pairs (a,b) with a taken modulo qm and b taken modulo n, with addition performed componentwise. The notation is used both for the additive group and, when context permits, for the corresponding ring direct product.

Properties and structure: Z_qm × Z_n is a finite abelian group of order qm·n. It is cyclic if and only if gcd(qm, n) = 1, in which case Z_qm × Z_n ≅ Z_{qm n}. In general, Z_qm × Z_n ≅ Z_g × Z_l, where g = gcd(qm, n) and l = lcm(qm, n). The order of an element (a,b) is the least common multiple of the orders of a in Z_qm and b in Z_n. Subgroups correspond to products of subgroups in each coordinate.

Examples: Z_4 × Z_6 is isomorphic to Z_2 × Z_12 (since gcd(4,6) = 2). Z_5 × Z_7 is cyclic of order 35, since gcd(5,7) = 1, and thus Z_5 × Z_7 ≅ Z_35.

Variants and applications: If Z_qm and Z_n are taken as rings, Z_qm × Z_n has a ring structure with componentwise addition and multiplication; it is not a domain or a field unless one component is trivial. By the Chinese Remainder Theorem, Z_{qm n} ≅ Z_qm × Z_n when gcd(qm, n) = 1. This construction is a standard example in the study of finite abelian groups and appears in coding theory, cryptography, and modular arithmetic.

See also: direct product, cyclic group, Chinese Remainder Theorem, invariant factor decomposition.