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.
As a subgroup
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

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