Defition of cyclic groups.

DgrayMan · **Joined:** Sat, 21 Jan 2012 03:59:22 UTC **Posts:** 182

I'm trying to understand the definition of a cyclic group:
"Cyclic Group: If 'a' is in a group G, and every element of G is a power of 'a' G={a^n : n in Z}, then G is a cyclic group and 'a' is its generator, G=<a>."

What does it mean by "every element of G is a power of 'a' ?" What is a example of this?

outermeasure · **Posted:** Sun, 8 Apr 2012 05:11:27 UTC

DgrayMan wrote:

I'm trying to understand the definition of a cyclic group:
"Cyclic Group: If 'a' is in a group G, and every element of G is a power of 'a' G={a^n : n in Z}, then G is a cyclic group and 'a' is its generator, G=<a>."

What does it mean by "every element of G is a power of 'a' ?" What is a example of this?

For example, take the group of rotations of a regular n-gon. The rotation about centre by $\frac{2\pi}{n}$ is a generator --- every other rotation is a composite of it or its inverse.

DgrayMan · **Joined:** Sat, 21 Jan 2012 03:59:22 UTC **Posts:** 182

I get it. i think typing it, actually helped me sound it out. thanks.

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Defition of cyclic groups.

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