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Concept
Center (group theory)
Summary
In abstract algebra, the center of a group, G, is the set of elements that commute with every element of G. It is denoted Z(G), from German Zentrum, meaning center. In set-builder notation,
Z(G) = .
The center is a normal subgroup, Z(G) ⊲ G. As a subgroup, it is always characteristic, but is not necessarily fully characteristic. The quotient group, G / Z(G), is isomorphic to the inner automorphism group, Inn(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.
The center of G is always a subgroup of G. In particular:
Z(G) contains the identity element of G, because it commutes with every element of g, by definition: eg = g = ge, where e is the identity;
If x and y are in Z(G), then so is xy, by associativity: (xy)g = x(yg) = x(gy) = (xg)y = (gx)y = g(xy) for each g ∈ G; i.e., Z(G) is closed;
If x is in Z(G), then so is x^−1 as, for all g in G, x^−1 commutes with g: (gx = xg) ⇒ (x^−1gxx^−1 = x^−1xgx^−1) ⇒ (x^−1g = gx^−1).
Furthermore, the center of G is always an abelian and normal subgroup of G. Since all elements of Z(G) commute, it is closed under conjugation.
Note that a homomorphism f: G → H between groups generally does not restrict to a homomorphism between their centers. Although f (Z (G)) commutes with f ( G ), unless f is surjective f (Z (G)) need not commute with all of H and therefore need not be a subset of Z ( H ). Put another way, there is no "center" functor between categories Grp and Ab. Even though we can map objects, we cannot map arrows.
By definition, the center is the set of elements for which the conjugacy class of each element is the element itself; i.e., Cl(g) = {g}.
The center is also the intersection of all the centralizers of each element of G. As centralizers are subgroups, this again shows that the center is a subgroup.
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
f(g)(h) = φ_g(h) = ghg^−1.
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