Brauer third main theorem
From Encyclopedia of Mathematics
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 
is defined to be the character 
such that 
for all 
(cf. also Character of a group). The block to which it belongs is called the principal block of the group algebra 
. The defect groups (cf. also Defect group of a block) of the principal block are the Sylow 
-subgroups of 
(cf. also 
-group). Let 
be a subgroup of 
, and let 
be a block of 
with defect group 
such that 
. Brauer's third main theorem states that 
is the principal block of 
if and only if 
is the principal block of 
.
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