# centralizer

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

$${C}_{G}(X)=\{g\in G:gx{g}^{-1}=x\mathit{\hspace{1em}}\text{for all}x\in X\}$$ |

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.

Title | centralizer |
---|---|

Canonical name | Centralizer1 |

Date of creation | 2013-03-22 14:01:20 |

Last modified on | 2013-03-22 14:01:20 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 8 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 58E40 |

Related topic | CentralizersInAlgebra |