# Center of a group

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) = {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 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).

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