# Thread: center vs normal

1. ## 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..

2. Let's call our group $\displaystyle G$.

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

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

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

A normal subgroup $\displaystyle N$ of $\displaystyle G$, denoted $\displaystyle N\trianglelefteq G$, is defined to be a subgroup with the property that, for any $\displaystyle n\in N$ and $\displaystyle g\in G$, the element $\displaystyle g^{-1}ng$ must be in $\displaystyle N$. 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, $\displaystyle Z(G)\trianglelefteq G$. This is easy to show:

For any $\displaystyle z\in Z(G),g\in G$, we have $\displaystyle g^{-1}zg=zg^{-1}g$ (because $\displaystyle z$ commutes with everything)
$\displaystyle =ze=z\in Z(G)$.

3. 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?

4. 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 $\displaystyle H$ is a subgroup of $\displaystyle G$, that means both that $\displaystyle H\subseteq G$ and that $\displaystyle H$ is a group in its own right.

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

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.