Cyclic Groups

Cyclic Group

A group is cyclic if that is, every has the form for some . We say that is generated by .

e.g. is cyclic, generated by either 1 or −1.

Proposition

If a group is cyclic, then it is abelian.

Proof This is straightforward from the definition.

Proposition

Let be a cyclic group. Then exactly one of the following holds:

1. is infinite
2. There exists such that .

Proof If is finite, To prove exists such that is equivalent to prove that for all , exists and such that . By the pigeonhole principle, for any integer exists that we have . Hence , so has a finite order . Let . Then for every , with and . It follows that $$ a^k=a^{n q+r}=\left(a^n\right)^q a^r=e^q a^r=a^r. $$$\square$

Let be a finite group and . Then . It follows that for all . In particular, if there exists such that then is cyclic.

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A group of prime order is cyclic.

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Theorem

A product of two finite cyclic groups with coprime orders is cyclic. i.e., if and only if .

The Structure Theorem for Finite Abelian Groups

Every finite abelian group is isomorphic to a direct product (sum) of cyclic groups: such that divides for all , and .

e.g.

Proposition

Let be a cyclic group of order . For each , there exists a unique subgroup of of order , and it is cyclic. Furthermore, the subgroups of are precisely the cyclic groups of order dividing .