# Math Help - Group order prime, center, abelian

1. ## Group order prime, center, abelian

Hello, I need help with this exercise

----- If #(G) = p^3, with p prime, then G is abelian or #(Z(G)) = p -----

THANKS!

Symmetric Group Center Trivial - ProofWiki

SORRY - ignore me! This is a completely different problem! Silly me.

3. Originally Posted by roporte
Hello, I need help with this exercise

----- If #(G) = p^3, with p prime, then G is abelian or #(Z(G)) = p -----

THANKS!
Burnisde's lemma says that $\text{Z}(G) = p,p^2,p^3$. If it is $p^3$ proof complete. Thus you need to show $\text{Z}(G) = p^2$ is impossible.