In abstract algebra, the center (or 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 characteristic subgroup of it.
If G is an abelian group then the center of G is all of G. At the other extreme, a group is said to be centerless 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 center of G and the image is called the inner automorphism group of G, denoted Inn(G). By the first isomorphism theorem G/Z(G) ≅ Inn(G).
See also: