In abstract algebra, the centre of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically,

Z(G) = {z ∈ G | gz = zg for all g ∈ G}

Note that Z(G) is a subgroup of G — if x and y are in Z(G), then for each g in G, (xy)g = x(yg) = x(gy) = (xg)y = (gx)y = g(xy) so xy is in Z(G) as well. A similar argument applies to inverses.

Moreover, Z(G) is an abelian subgroup of G, a normal subgroup of G, and even a strictly characteristic subgroup of G, but not always fully characteristic.

The centre of G is all of G iff G is an abelian group. At the other extreme, a group is said to be centreless if Z(G) is trivial.

Consider the map f: G → Aut(G) to the automorphism group of G defined by f(g)(h) = ghg⁻¹. The kernel of this map is the centre of G and the image is called the inner automorphism group of G, denoted Int(G). By the first isomorphism theorem G/Z(G) $\backslash cong$ Int(G).

Examples

• The centre of the orthogonal group O(n ) is { I, −I }.
• The centre of the quaternion group Q = {1, −1, i, −i, j, −j, k, −k} is {1, −1}.
• Using the class equation one can prove that the centre of a finite p-group is non-trivial

See also

• centre (algebra)
• centralizer and normalizer
• conjugacy class.

Group theory | Functional subgroups

Zentrum (Gruppentheorie) | Centre d'un groupe | Centro di un gruppo | מרכז של חבורה | Centrum (algebra) | 中心 (群论)