# center vs normal

• Feb 11th 2011, 03:47 PM
alexandrabel90
center vs normal
May i know what is the difference between the normal subgroup of a group and the centre of a group?

from what i know, both of them contain elements that commute with all elements in the group..
• Feb 11th 2011, 05:10 PM
topspin1617
Let's call our group $G$.

The center of a group, usually denoted $Z(G)$, is exactly the set of elements that commute with every other element in $G$:

$Z(G)=\{g\in G|gx=xg\forall x\in G\}$
.

It is NOT true that all elements of an arbitrary normal subgroup commute with every element of $G$.

A normal subgroup $N$ of $G$, denoted $N\trianglelefteq G$, is defined to be a subgroup with the property that, for any $n\in N$ and $g\in G$, the element $g^{-1}ng$ must be in $N$. At first glance it seems to be a strange property, but it turns out to be a very important one.

It IS true, however, that the center of a group is always a normal subgroup of the group; that is, $Z(G)\trianglelefteq G$. This is easy to show:

For any $z\in Z(G),g\in G$, we have $g^{-1}zg=zg^{-1}g$ (because $z$ commutes with everything)
$=ze=z\in Z(G)$.
• Feb 11th 2011, 05:21 PM
alexandrabel90
does that mean that the centre of the group has the added property that it is abelian? whereas the normal of a group might not be abelian?
• Feb 11th 2011, 05:35 PM
topspin1617
Quote:

Originally Posted by alexandrabel90
does that mean that the centre of the group has the added property that it is abelian? whereas the normal of a group might not be abelian?

Yes. Recall that when we say something like $H$ is a subgroup of $G$, that means both that $H\subseteq G$ and that $H$ is a group in its own right.

$Z(G)$ is most certainly an abelian group; the elements of $Z(G)$ commute with any element in the entire group, which of course include the elements of $Z(G)$ itself (because $Z(G)\subseteq G$).

But an arbitrary normal subgroup of some group need not be abelian; it's definitely possible for such a subgroup to be abelian, but it just depends on the situation. It is in no means a necessary condition for normality.