Brauer third main theorem
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) |
Brauer third main theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brauer_third_main_theorem&oldid=32310