Group order prime, center, abelian

**roporte** · September 13th 2008, 12:13 AM

Hello, I need help with this exercise

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

THANKS!

**Matt Westwood** · September 13th 2008, 01:33 AM

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.

**ThePerfectHacker** · September 13th 2008, 05:31 PM

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.

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