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Centralizer


The centralizer of an element z of a group G is the set of elements of G which commute with z,

 C_G(z)={x in G,xz=zx}.

Likewise, the centralizer of a subgroup H of a group G is the set of elements of G which commute with every element of H,

 C_G(H)={x in G, forall h in H,xh=hx}.

The centralizer always contains the group center of the group and is contained in the corresponding normalizer. In an Abelian group, the centralizer is the whole group.


See also

Abelian Group, Group, Group Center, Normalizer, Subgroup

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Cite this as:

Weisstein, Eric W. "Centralizer." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Centralizer.html

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