The Center of a Group
The center of a groupwrittenis the set of elementsthat satisfyfor all– that this, they commute with all
The center is a subgroup ofand is abelian, since every element ofcommutes with every element ofso also commutes with every element ofWe can prove thatis a subgroup ofusing the subgroup axioms.
S1:implyIfthenandfor allso
S2:for allso
S3:impliesimpliesfor allso
As a subgroup,is normal sincefor allsoand The quotient groupis well defined.
A groupis abelian if and only ifAt the other extreme, a group is said to be centerless if
Consider the map
This is a group homomorphism, and its kernel is preciselysince ifIts image is called the inner automorphism group ofdenoted
Examples

The center of a nonabelian simple group is trivial, since simple groups have no nontrivial normal subgroups.

The center of the dihedral groupis trivial whenis odd. Whenis even, the center consists of the identity element together with the half rotation.

The center of the quaternion groupis

The center of the symmetric groupis trivial for

The center of the alternating groupis trivial for

The center of the general linear groupis the collection of scalar matrices

The center of the orthogonal groupis

The center of the multiplicative group of nonzero quaternions is the multiplicative group of nonzero real numbers.

Using the class equation one can prove that the center of any nontrivial finite pgroup is nontrivial.

If the quotient group G / Z(G) is cyclic, G is abelian.