Review of Group Theory: The Center of a Group
Point of post: In this post we take a brief break from our discussion of group actions to address the notion of the center of a group.
Just as was the case for the center of an algebra it is often fruitful to consider the center of a group. In essence, the center of a group is merely the set of all elements of the group which commute with all other elements of the group. We shall use the concepts in this post in this and the next post to conclude that for any group of order with prime must be abelian.
The Center of a Group
Let be a group, we define the center of to be
Our first theorem says that the center of a group is a normal subgroup of a group:
Theorem: Let be a group, then . Moreover, (where is the set of inner automorphisms).
Proof: Recall that is a homomorphism. Note then that
from where the conclusion follows from our earlier characterization of normality. The fact that now follows immediately from the First Isomorphism Theorem..
An important theorem involving the center of a group is the following
Theorem: Let be a group. Then, if is cyclic then is abelian.
Proof: Let . Then, for any we have that there exists such that and . Consequently there exists such that and . Thus,
the conclusion follows.
The reason why this is slightly important is that it well let us conclude (later) that the only groups up to isomorphism of order where is prime are or where is a symbol for the generic cyclic group of order and is the direct product of groups which we have yet to discuss.
1. Lang, Serge. Undergraduate Algebra. 3rd. ed. Springer, 2010. Print.
2. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 200