Abelian group
Finite abelian groups

Cyclic groups of integers modulo n {\displaystyle n}, Z / n Z {\displaystyle \mathbb {Z} /n\mathbb {Z} }, were among the first examples of groups. It turns out that an arbitrary finite abelian group is isomorphic to a direct sum of finite cyclic groups of prime power order, and these orders are uniquely determined, forming a complete system of invariants. The automorphism group of a finite abelian group can be described directly in terms of these invariants. The theory had been first developed in the 1879 paper of Georg Frobenius and Ludwig Stickelberger and later was both simplified and generalized to finitely generated modules over a principal ideal domain, forming an important chapter of linear algebra.

Any group of prime order is isomorphic to a cyclic group and therefore abelian. Any group whose order is a square of a prime number is also abelian. In fact, for every prime number p {\displaystyle p} there are (up to isomorphism) exactly two groups of order p 2 {\displaystyle p^{2}}, namely Z p 2 {\displaystyle \mathbb {Z} _{p^{2}}} and Z p × Z p {\displaystyle \mathbb {Z} _{p}\times \mathbb {Z} _{p}}.
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