Center
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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) = {z ∈ G ∣ ∀g ∈ G, zg = gz}
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.