View source for Brauer second main theorem

For notation and definitions, see [[Brauer first main theorem|Brauer first main theorem]].

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b1204601.png" /> be an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b1204602.png" /> whose order is a power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b1204603.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b1204605.png" />-section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b1204606.png" /> associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b1204607.png" /> is the set of all elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b1204608.png" /> whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046010.png" />-part is conjugate to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046011.png" />. Brauer second main theorem relates the values of irreducible characters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046012.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046013.png" />-section associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046014.png" /> to values of characters in certain blocks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046015.png" />.

Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046016.png" /> is an irreducible character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046017.png" /> (cf. also [[Character of a group|Character of a group]]), afforded by the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046018.png" />-free right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046019.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046020.png" />, and belonging to the [[Block|block]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046021.png" /> (cf. also [[Defect group of a block|Defect group of a block]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046022.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046023.png" />-element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046024.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046025.png" />. For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046026.png" />-subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046027.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046029.png" />; hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046030.png" /> is defined for all blocks <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046031.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046032.png" />. One can organize the block decomposition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046033.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046034.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046035.png" /> be the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046036.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046037.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046038.png" /> be the projection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046039.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046040.png" />. The restriction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046041.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046042.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046043.png" /> can be decomposed as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046044.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046045.png" /> is the character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046047.png" /> is the character of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046048.png" />, then of course <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046049.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046050.png" />. Brauer's second main theorem states that for all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046051.png" /> of order prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046053.png" />. Thus, the values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046054.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046055.png" />-section associated to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046056.png" /> are determined in the blocks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046057.png" /> sent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120460/b12046058.png" /> by the Brauer correspondence (cf. also [[Brauer first main theorem|Brauer first main theorem]]).

This theorem was first proved in [[#References|[a1]]]. See also [[#References|[a2]]], [[#References|[a3]]], and [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Brauer,   "Zur Darstellungstheorie der Gruppen endlicher Ordnung II"  ''Math. Z.'' , '''72'''  (1959)  pp. 22–46</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Curtis,   I. Reiner,   "Methods of representation theory" , '''II''' , Wiley  (1987)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Feit,   "The representation theory of finite groups" , North-Holland  (1982)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Nagao,   Y. Tsushima,   "Representation of finite groups" , Acad. Press  (1987)</TD></TR></table>