Center (group theory)
Set of elements that commute with every element of a group / From Wikipedia, the free encyclopedia
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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}.
"Group center" redirects here. For the American counter-cultural group, see Aldo Tambellini § Lower East Side artists.
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The center is a normal subgroup, Z(G) ⊲ G, and also a characteristic subgroup, 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 central elements.