Center (group theory)

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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. That is,

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

Note that Z(G) is a subgroup of G, because

1. Z(G) contains e, the identity element, because e ∈ G and eg = ge for all g ∈ G by definition of e, so by definition of Z(G), e ∈ Z(G);
2. If x and y are in Z(G), then (xy)g = x(yg) = x(gy) = (xg)y = (gx)y = g(xy) for each g in G, and so xy is in Z(G) as well (i.e., Z(G) exhibits closure);
3. If x is in Z(G), then gx = xg, and multiplying twice, once on the left and once on the right, by x⁻¹, gives x⁻¹g = gx⁻¹ — so x⁻¹ ∈ Z(G).

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 if and only if 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) from G to the automorphism group of G defined by f(g) = φ_g, where φ_g is the automorphism of G defined by φ_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).

[edit] Examples

• The center of the invertible matrices GL_n(F) (F any field) is the collection of scalar matrices .
• The center of the orthogonal group O(n,F) is {I_n, − I_n}.
• The center 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 center of a non-trivial finite p-group is non-trivial

[edit] See also

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