In abstract algebra, the center of a group G, denoted Z(G),^{[note 1]} is the set of elements that commute with every element of G. In setbuilder notation,
The center is a subgroup of G, which by definition is abelian (that is commutative). As a subgroup, it is always normal, and indeed characteristic, but it need not be fully characteristic. The quotient group G / Z(G) is isomorphic to the group of inner automorphisms of G.
A group G is abelian if and only if Z(G) = G. At the other extreme, a group is said to be centerless if Z(G) is trivial, i.e. consists only of the identity element.
The elements of the center are sometimes called central.
Contents
As a subgroup
The center of G is always a subgroup of G. In particular:
Furthermore the center of G is always a normal subgroup of G, as it is closed under conjugation.
Conjugation
Consider the map f: G → Aut(G) from G to the automorphism group of G defined by f(g) = φ_{g}, where φ_{g} is the automorphism of G defined by
This is a group homomorphism, and its kernel is precisely the center of G, and its image is called the inner automorphism group of G, denoted Inn(G). By the first isomorphism theorem we get
The cokernel of this map is the group of outer automorphisms, and these form the exact sequence
Examples
 The center of an abelian group G is all of G.
 The center of a nonabelian simple group is trivial.
 The center of the dihedral group D_{n} is trivial when n is odd. When n is even, the center consists of the identity element together with the 180° rotation of the polygon.
 The center of the quaternion group Q_{8} = {1, − 1,i, − i,j, − j,k, − k} is {1, − 1}.
 The center of the symmetric group S_{n} is trivial for n ≥ 3.
 The center of the alternating group A_{n} is trivial for n ≥ 5.
 The center of the general linear group GL_{n}(F) is the collection of scalar matrices .
 The center of the orthogonal group O(n,F) is {I_{n}, − I_{n}}.
 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.
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