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  "body": "\n## Summary\n\nIn mathematics, especially group theory, the **centralizer** (also called **commutant**) of a [[subset|0BJR9QR08E]] *S* in a group *G* is the set $\\operatorname{C}_G(S)$ of elements of *G* that commute with every element of *S*, or equivalently, the set of elements $g\\in G$ such that conjugation by $g$ leaves each element of *S* fixed. The **normalizer** of *S* in *G* is the [[set|0BSJ7RQ9BW]] of elements $\\mathrm{N}_G(S)$ of *G* that satisfy the weaker condition of leaving the set $S \\subseteq G$ fixed under conjugation. The centralizer and normalizer of *S* are [[subgroups|0X1JXT47GA]] of *G*. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets *S*.\n",
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