Brauer's three main theorems

Brauer's main theorems are three theorems in representation theory of finite groups linking the blocks of a finite group (in characteristic p) with those of its p-local subgroups, that is to say, the normalizers of its nontrivial p-subgroups.

The second and third main theorems allow refinements of orthogonality relations for ordinary characters which may be applied in finite group theory. These do not presently admit a proof purely in terms of ordinary characters. All three main theorems are stated in terms of the Brauer correspondence.

Brauer correspondence

There are many ways to extend the definition which follows, but this is close to the early treatments by Brauer. Let G be a finite group, p be a prime, F be a field of characteristic p. Let H be a subgroup of G which contains

$QC_{G}(Q)$

for some p-subgroup Q of G, and is contained in the normalizer

$N_{G}(Q)$,

where $C_{G}(Q)$ is the centralizer of Q in G.

The Brauer homomorphism (with respect to H) is a linear map from the center of the group algebra of G over F to the corresponding algebra for H. Specifically, it is the restriction to $Z(FG)$ of the (linear) projection from $FG$ to $FC_{G}(Q)$ whose kernel is spanned by the elements of G outside $C_{G}(Q)$. The image of this map is contained in $Z(FH)$, and it transpires that the map is also a ring homomorphism.

Since it is a ring homomorphism, for any block B of FG, the Brauer homomorphism sends the identity element of B either to 0 or to an idempotent element. In the latter case, the idempotent may be decomposed as a sum of (mutually orthogonal) primitive idempotents of Z(FH). Each of these primitive idempotents is the multiplicative identity of some block of FH. The block b of FH is said to be a Brauer correspondent of B if its identity element occurs in this decomposition of the image of the identity of B under the Brauer homomorphism.

Brauer's first main theorem

Brauer's first main theorem (Brauer 1944, 1956, 1970) states that if $G$ is a finite group and $D$ is a $p$-subgroup of $G$, then there is a bijection between the set of (characteristic p) blocks of $G$ with defect group $D$ and blocks of the normalizer $N_{G}(D)$ with defect group D. This bijection arises because when $H=N_{G}(D)$, each block of G with defect group D has a unique Brauer correspondent block of H, which also has defect group D.

Brauer's second main theorem

Brauer's second main theorem (Brauer 1944, 1959) gives, for an element t whose order is a power of a prime p, a criterion for a (characteristic p) block of $C_{G}(t)$ to correspond to a given block of $G$, via generalized decomposition numbers. These are the coefficients which occur when the restrictions of ordinary characters of $G$ (from the given block) to elements of the form tu, where u ranges over elements of order prime to p in $C_{G}(t)$, are written as linear combinations of the irreducible Brauer characters of $C_{G}(t)$. The content of the theorem is that it is only necessary to use Brauer characters from blocks of $C_{G}(t)$ which are Brauer correspondents of the chosen block of G.

Brauer's third main theorem

Brauer's third main theorem (Brauer 1964, theorem3) states that when Q is a p-subgroup of the finite group G, and H is a subgroup of G containing $QC_{G}(Q)$ and contained in $N_{G}(Q)$, then the principal block of H is the only Brauer correspondent of the principal block of G (where the blocks referred to are calculated in characteristic p).

References

• Brauer, R. (1944), "On the arithmetic in a group ring", Proceedings of the National Academy of Sciences of the United States of America, 30 (5): 109–114, doi:10.1073/pnas.30.5.109, ISSN 0027-8424, JSTOR 87919, MR 0010547, PMC 1078679, PMID 16578120
• Brauer, R. (1946), "On blocks of characters of groups of finite order I", Proceedings of the National Academy of Sciences of the United States of America, 32 (6): 182–186, doi:10.1073/pnas.32.6.182, ISSN 0027-8424, JSTOR 87578, MR 0016418, PMC 1078910, PMID 16578199
• Brauer, R. (1946), "On blocks of characters of groups of finite order. II", Proceedings of the National Academy of Sciences of the United States of America, 32 (8): 215–219, doi:10.1073/pnas.32.8.215, ISSN 0027-8424, JSTOR 87838, MR 0017280, PMC 1078924, PMID 16578207
• Brauer, R. (1956), "Zur Darstellungstheorie der Gruppen endlicher Ordnung", Mathematische Zeitschrift, 63: 406–444, doi:10.1007/BF01187950, ISSN 0025-5874, MR 0075953, S2CID 186223039
• Brauer, R. (1959), "Zur Darstellungstheorie der Gruppen endlicher Ordnung. II", Mathematische Zeitschrift, 72: 25–46, doi:10.1007/BF01162934, ISSN 0025-5874, MR 0108542, S2CID 122127627
• Brauer, R. (1964), "Some applications of the theory of blocks of characters of finite groups. I", Journal of Algebra, 1 (2): 152–167, doi:10.1016/0021-8693(64)90031-6, ISSN 0021-8693, MR 0168662
• Brauer, R. (1970), "On the first main theorem on blocks of characters of finite groups.", Illinois Journal of Mathematics, 14 (2): 183–187, doi:10.1215/ijm/1256053174, ISSN 0019-2082, MR 0267010
• Dade, Everett C. (1971), "Character theory pertaining to finite simple groups", in Powell, M. B.; Higman, Graham (eds.), Finite simple groups. Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969., Boston, MA: Academic Press, pp. 249–327, ISBN 978-0-12-563850-0, MR 0360785 gives a detailed proof of the Brauer's main theorems.
• Ellers, H. (2001) [1994], "Brauer's first main theorem", Encyclopedia of Mathematics, EMS Press
• Ellers, H. (2001) [1994], "Brauer height-zero conjecture", Encyclopedia of Mathematics, EMS Press
• Ellers, H. (2001) [1994], "Brauer's second main theorem", Encyclopedia of Mathematics, EMS Press
• Ellers, H. (2001) [1994], "Brauer's third main theorem", Encyclopedia of Mathematics, EMS Press
• Walter Feit, The representation theory of finite groups. North-Holland Mathematical Library, 25. North-Holland Publishing Co., Amsterdam-New York, 1982. xiv+502 pp. ISBN 0-444-86155-6