# Group order prime, center, abelian

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• September 13th 2008, 12:13 AM
roporte
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!
• September 13th 2008, 01:33 AM
Matt Westwood
This one comes up again and again. Try this page:

Symmetric Group Center Trivial - ProofWiki

SORRY - ignore me! This is a completely different problem! Silly me.
• September 13th 2008, 05:31 PM
ThePerfectHacker
Quote:

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.