Difference between revisions of "Brauer third main theorem"
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For notation and definitions, see [[Brauer first main theorem|Brauer first main theorem]]. | For notation and definitions, see [[Brauer first main theorem|Brauer first main theorem]]. | ||
| − | Brauer's third main theorem deals with one situation in which the Brauer correspondence (cf. also [[Brauer first main theorem|Brauer first main theorem]]) is easy to compute. The principal character of a [[Group|group]] | + | Brauer's third main theorem deals with one situation in which the Brauer correspondence (cf. also [[Brauer first main theorem|Brauer first main theorem]]) is easy to compute. The principal character of a [[Group|group]] $G$ is defined to be the character $\chi$ such that $\chi(g)=1$ for all $g\in G$ (cf. also [[Character of a group|Character of a group]]). The block to which it belongs is called the principal block of the [[Group algebra|group algebra]] $RG$. The defect groups (cf. also [[Defect group of a block|Defect group of a block]]) of the principal block are the Sylow $p$-subgroups of $G$ (cf. also [[P-group|$p$-group]]). Let $H$ be a subgroup of $G$, and let $b$ be a block of $H$ with defect group $D$ such that $C_G(D)\subseteq H$. Brauer's third main theorem states that $b^G$ is the principal block of $RG$ if and only if $b$ is the principal block of $RH$. |
See [[#References|[a1]]], [[#References|[a2]]], and [[#References|[a3]]]. | See [[#References|[a1]]], [[#References|[a2]]], and [[#References|[a3]]]. | ||
Latest revision as of 09:15, 27 June 2014
For notation and definitions, see Brauer first main theorem.
Brauer's third main theorem deals with one situation in which the Brauer correspondence (cf. also Brauer first main theorem) is easy to compute. The principal character of a group $G$ is defined to be the character $\chi$ such that $\chi(g)=1$ for all $g\in G$ (cf. also Character of a group). The block to which it belongs is called the principal block of the group algebra $RG$. The defect groups (cf. also Defect group of a block) of the principal block are the Sylow $p$-subgroups of $G$ (cf. also $p$-group). Let $H$ be a subgroup of $G$, and let $b$ be a block of $H$ with defect group $D$ such that $C_G(D)\subseteq H$. Brauer's third main theorem states that $b^G$ is the principal block of $RG$ if and only if $b$ is the principal block of $RH$.
References
| [a1] | J.L. Alperin, "Local representation theory" , Cambridge Univ. Press (1986) |
| [a2] | C. Curtis, I. Reiner, "Methods of representation theory" , II , Wiley (1987) |
| [a3] | H. Nagao, Y. Tsushima, "Representation of finite groups" , Acad. Press (1987) |
How to Cite This Entry:
Brauer third main theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brauer_third_main_theorem&oldid=11668
Brauer third main theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brauer_third_main_theorem&oldid=11668
This article was adapted from an original article by H. Ellers (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article