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..
Let's call our group .
The center of a group, usually denoted , is exactly the set of elements that commute with every other element in :
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It is NOT true that all elements of an arbitrary normal subgroup commute with every element of .
A normal subgroup of , denoted , is defined to be a subgroup with the property that, for any and , the element must be in . 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, . This is easy to show:
For any , we have (because commutes with everything)
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Yes. Recall that when we say something like is a subgroup of , that means both that and that is a group in its own right.
is most certainly an abelian group; the elements of commute with any element in the entire group, which of course include the elements of itself (because ).
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.