Definition of cyclic group: let it be an element of a group, then the subgroup formed by the set of all powers of is called cyclic subgroup generated by, and it is recorded as. If a group happens to be generated by one of its elements, that is, it exists, then this group is called a cyclic group and is the generator of this group. Theorem: Every cyclic group is an Abel group.

Any group must have cyclic subgroups, and the concept of element period is not limited to cyclic groups. As long as it is an element in a group, it has its own elements.

In addition, let G be a group, and if there is a ∈G, let.

G= (a)= {ak

k∈Z}

Then g is called a cyclic group, and a is the generator of g.

O if g is an infinite set, it is called an infinite cyclic group;

O if g has n elements, i.e. |G|=n, then g is called a cyclic group of order n.

Cyclic groups are commutative groups.

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