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..

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- Feb 11th 2011, 03:47 PMalexandrabel90center 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 PMtopspin1617
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)

$\displaystyle =ze=z\in Z(G)$. - Feb 11th 2011, 05:21 PMalexandrabel90
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 PMtopspin1617
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.