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centralizer (Definition)

Let $ G$ a group acting on itself by conjugation. Let $ X$ be a subset of $ G$. The stabilizer of $ X$ is called the centralizer of $ X$ and it's the set

$\displaystyle C_G(X)=\{g\in G : gxg^{-1}=x$   for all $\displaystyle x\in X\}$

For any group $ G$, $ C_G(G)=Z(G)$, the center of $ G$. Thus, any subgroup of $ C_G(G)$ is an abelian subgroup of $ G$. However, the converse is generally not true. For example, take any non-abelian group and pick any element not in the center. Then the subgroup generated by it is obviously abelian, clearly non-trivial and not contained in the center.



"centralizer" is owned by yark. [ full author list (3) | owner history (4) ]
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See Also: centralizers in algebra

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Cross-references: contained, subgroup generated by, non-abelian group, converse, abelian, subgroup, center, stabilizer, subset, conjugation, group
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This is version 5 of centralizer, born on 2003-10-15, modified 2008-01-27.
Object id is 4978, canonical name is CentralizerOfASubsetOfAGroup.
Accessed 1919 times total.

Classification:
AMS MSC58E40 (Global analysis, analysis on manifolds :: Variational problems in infinite-dimensional spaces :: Group actions)

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