Prove G is Cyclic: Prime p Order Group

[catherinenanc]

1. Let p be a prime and G a group whose order is p. Prove that G is cyclic.



2. I know that if p is prime, then the only possible subgroups of G are {e} and G itself. But, how to use this fact to show that G is cyclic?

 

[Dick]

Pick an element g of G that is not e and consider the subgroup generated by g.

 

[Karamata]

HINT: Lagrange

 

[catherinenanc]

Pick an element g of G that is not e and consider the subgroup generated by g.

Ok, this may sound stupid, but, how do I know that <g>=G?

 

[catherinenanc]

Ok, this may sound stupid, but, how do I know that <g>=G?

Oh, wait, there are only two subgroups,so it has to be.

Thanks!