Proving Cyclic Groups: A Step-by-Step Guide

[Fairy111]

Homework Statement

How do i go about proving that a group is cyclic?

Homework Equations

The Attempt at a Solution

 

[dx]

Start with the definition of a cyclic group, and see if your group satisfies it.

 

[Fairy111]

The group, G, is a finte group with cardinality p, a prime integer. How should i start off, if i need to prove it's cyclic?

 

[dx]

Are you familiar with Lagrange's theorem?

 

[Fairy111]

yes, i know that the only subgroups of G are itself and the subgroup {e} which consists of the neutral element. This is because the only possibilities of the cardinalities of the subgroups are 1 or p.

 

[dx]

Ok, now pick an element of the group G, say g not equal to 1. What are the possible orders of g?

 

[Fairy111]

possible orders of g are 1 or p? Since those are the only numbers that divide the prime number p.

 

[dx]

It cannot be 1 because we assumed g was not equal to the identity. So the order of g must be p, and therefore G = {1 , g, g², ... , g^p-1} which is cyclic.

 

[Fairy111]

Ok, thanku very much for the help:)

 

[dx]

No problem.