# Thread: Cardinality of the center of a group

1. ## Cardinality of the center of a group

Problem:

Let G be a group such that $\displaystyle \left| G \right| = {p^3}$. Show that the center of G has more than one element, i.e. $\displaystyle \left| {Z\left( G \right)} \right| > 1$.

[I'm wondering how to solve this. I'm supposing this involves the Sylow theorems but I don't know how to apply them in this case. Any hint or technique will help a lot.]

2. Originally Posted by guildmage
Problem:

Let G be a group such that $\displaystyle \left| G \right| = {p^3}$. Show that the center of G has more than one element, i.e. $\displaystyle \left| {Z\left( G \right)} \right| > 1$.

[I'm wondering how to solve this. I'm supposing this involves the Sylow theorems but I don't know how to apply them in this case. Any hint or technique will help a lot.]

Sylow theorems will hardly help here since the group is a p-group...You need to know the class equation, which follows from the action of the group on itself by conjugation, and then it is almost immediate not only that the center of a finite p-group is non-trivial but in fact it has at least p elements. Any decent group theory book talks about this.

Tonio