Center (group Theory)
In abstract algebra, the center of a group G, denoted Z(G), is the set of elements that commute with every element of G. In set-builder notation,
- .
The center is a subgroup of G, which by definition is abelian (that is commutative). As a subgroup, it is always normal, and indeed characteristic, but it need not be fully characteristic. The quotient group G / Z(G) is isomorphic to the group of inner automorphisms of 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.
Read more about Center (group Theory): As A Subgroup, Conjugacy Classes and Centralisers, Conjugation, Examples, Higher Centers
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