Let's call our group .

Thecenterof a group, usually denoted , is exactly the set of elements that commute with every other element in :

.

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

Anormalsubgroup 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)

.