# Analysis of Cyclic Groups

Describe what cyclic groups are and show that they are abelian. Describe their structure and the structure of their subgroups and factor groups up to isomorphism.

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#### Solution Preview

A cyclic group is generated by an element of the group. That means that every element of the group is a power of some element of the group, referred to as the generator.

The order of the group is the number of members, or elements, of the group. So, a cyclic group of order 5 may be written as the set {g^0,g, g^2, g^3, g^4} under an appropriate binary operation. g^5 = g^0 which is the identity element.

Any cyclic group of order n ...

#### Solution Summary

Cyclic groups and their relationship to the group of integers under the operation addition is explained in detail, including looking at the structure of their subgroups and factor (or quotient) groups.