## The Center of a Group

The center of a group written is the set of elements that satisfy for all – that this, they commute with all The center is a subgroup of and is abelian, since every element of commutes with every element of so also commutes with every element of We can prove that is a subgroup of using the subgroup axioms.

S1: imply If then and for all  so S2:  for all so S3: implies  implies for all so As a subgroup, is normal since for all so and The quotient group is well defined.

A group is abelian if and only if At the other extreme, a group is said to be centerless if Consider the map This is a group homomorphism, and its kernel is precisely since if Its image is called the inner automorphism group of denoted ### Examples

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

• The center of the dihedral group is trivial when is odd. When is even, the center consists of the identity element together with the half rotation.

• The center of the quaternion group is • The center of the symmetric group is trivial for • The center of the alternating group is trivial for • The center of the general linear group is the collection of scalar matrices • The center of the orthogonal group is • The center of the multiplicative group of non-zero quaternions is the multiplicative group of non-zero real numbers.

• Using the class equation one can prove that the center of any non-trivial finite p-group is non-trivial.

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