- #1

jostpuur

- 2,116

- 19

## Homework Statement

Suppose [itex]p[/itex] is some prime number, and [itex]G[/itex] a group such that [itex]\# G = p^n[/itex] with some [itex]n\in\{1,2,3,\ldots\}[/itex]. Prove that the center

[tex]

Z(G) = \{g\in G\;|\; gg'=g'g\;\forall g'\in G\}

[/tex]

contains more than a one element.

## Homework Equations

Obviously [itex]1\in Z(G)[/itex], so the task is to find some other element from there too.

A hint is given, that conjugacy classes

[tex]

[x]=\{x'\in G\;|\; \exists y\in G,\; x'=yxy^{-1}\}

[/tex]

are supposed to be examined.

## The Attempt at a Solution

Nothing to be done in sight.

I have some results concerning Sylow p-subgroups, but I don't see how they could be used.