# centralizer

Let $G$ be a group. The centralizer^{} of an element $a\in G$ is defined to be the set

$$C(a)=\{x\in G\mid xa=ax\}$$ |

Observe that, by definition, $e\in C(a)$, and that if $x,y\in C(a)$, then $x{y}^{-1}a=x{y}^{-1}a(y{y}^{-1})=x{y}^{-1}ya{y}^{-1}=xa{y}^{-1}=ax{y}^{-1}$, so that $x{y}^{-1}\in C(a)$. Thus $C(a)$ is a subgroup^{} of $G$. For $a\ne e$, the subgroup is non-trivial, containing at least $\{e,a\}$.

To illustrate an application of this concept we prove the following lemma.

Lemma:

There exists a bijection between the right cosets^{} of $C(a)$ and the conjugates^{} of $a$.

Proof:

If $x,y\in G$ are in the same right coset, then $y=cx$ for some $c\in C(a)$. Thus ${y}^{-1}ay={x}^{-1}{c}^{-1}acx={x}^{-1}{c}^{-1}cax={x}^{-1}ax$.
Conversely, if ${y}^{-1}ay={x}^{-1}ax$ then $x{y}^{-1}a=ax{y}^{-1}$ and $x{y}^{-1}\in C(a)$ giving $x,y$ are in the same right coset.
Let $[a]$ denote the conjugacy class^{} of $a$. It follows that $|[a]|=[G:C(a)]$ and $|[a]|\mid |G|$.

We remark that $a\in Z(G)\iff C(a)=G\iff |[a]|=1$, where $Z(G)$ denotes the center of $G$.

Now let $G$ be a $p$-group, i.e. a finite group^{} of order ${p}^{n}$,
where $p$ is a prime and $n$ is a positive integer.
Let $z=|Z(G)|$.
Summing over elements in distinct conjugacy classes,
we have ${p}^{n}=\sum |[a]|=z+{\sum}_{a\notin Z(G)}|[a]|$
since the center consists precisely of the conjugacy classes of
cardinality $1$.
But $|[a]|\mid {p}^{n}$, so $p\mid z$.
However, $Z(G)$ is certainly non-empty, so we conclude that every
$p$-group has a non-trivial center.

The groups $C(ga{g}^{-1})$ and $C(a)$, for any $g$, are isomorphic^{}.

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

Canonical name | Centralizer |

Date of creation | 2013-03-22 12:35:01 |

Last modified on | 2013-03-22 12:35:01 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 14 |

Author | drini (3) |

Entry type | Definition |

Classification | msc 20-00 |

Synonym | centraliser |

Related topic | Normalizer^{} |

Related topic | GroupCentre |

Related topic | ClassEquationTheorem |