# Finite abelian groups with elements of certain orders

Give examples of finite abelian groups in which all elements (except the identity element) are of the same order.

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#### Solution Preview

Suppose that n > 1, and we are asked to name an abelian group all of whose elements are of order n. Technically, no such group exists, since the identity element of every group (abelian or otherwise) is of order 1.

So suppose we try to find finite abelian groups such that all the elements except the identity element are of the same order.

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One important class of abelian groups consists of the cyclic groups (the groups that can be generated from a single element).

If n > 1 and n is the order of the cyclic group of order n, then every element is of order m for some ...

#### Solution Summary

Examples are given of finite abelian groups in which all elements but the identity element are of the same order. The reason why they have the same order is explained.