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A cyclic group of order n is isomorphic to the additive group ℤ/nℤ.
Proposition (finite cyclic groups). Let be a group. Suppose is cyclic of finite order . Then is isomorphic to the additive group . Concretely, the chosen generator determines the isomorphism
is a well-defined isomorphism.
This identifies finite cyclic groups up to isomorphism by their order. The displayed isomorphism is not canonical: it depends on the choice of generator .
A group is a set together with a binary operation such that:
Equivalently, a group is a monoid in which every element is invertible. Much of group theory studies subgroups and structure-preserving maps called group homomorphisms.
Let be groups. A group isomorphism is a group homomorphism that is bijective as a bijective function.
If is an isomorphism, then its inverse function is also a group homomorphism, so gives a structure-preserving identification between and . One writes to denote that there exists a group isomorphism between them.