Z2Zntype

Z2Zntype is a designation used in algebra to refer to a class of finite abelian groups that are isomorphic to the direct product Z_2 × Z_n for some integer n ≥ 2. The term highlights the presence of two invariant factors, with the first factor of order 2, and it is commonly encountered in discussions of the invariant factor decomposition of finite abelian groups.

The group Z_2 × Z_n has order 2n. If n is odd, gcd(2,n) = 1, and Z_2 × Z_n is isomorphic to the cyclic group Z_{2n}. If n is even, the product is noncyclic, giving a nontrivial example of a two-generator, finite abelian group. In the standard invariant-factor framework, Z_2 × Z_n fits the form Z_{d1} × Z_{d2} with d1 dividing d2 and d1 = 2, illustrating two-factor structure with a fixed first invariant factor.

Properties of Z2Zntype groups include their abelian and finite nature, and their subgroup structure reflects the two-factor decomposition. The 2-primary component and the overall automorphism group vary with n, offering a straightforward setting to study how the presence of a fixed 2-factor influences these aspects.

Examples help illustrate the distinction between cyclic and noncyclic cases: when n = 3, Z_2 × Z_3 ≅ Z_6 is cyclic; when n = 4, Z_2 × Z_4 has order 8 and is noncyclic. Z2Zntype thus serves as a simple, concrete family used in teaching and in exploring invariant-factor classifications and two-generator abelian groups.

See also: finite abelian group, invariant factor decomposition, direct product, Z_n, Z_2.