# Center (group theory)

(Redirected from Center of a group)

In abstract algebra, the center 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) = {zG | gz = zg for all gG}

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 center of G is all of G iff G is an abelian group. 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) $\displaystyle \cong$ Inn(G).

Example:

The center of the orthogonal group O(n ) is { I, −I }.