Brauer's three main theorems

{{short description|Three results in the representation theory of finite groups}}

{{update|date=October 2024|reason=https://www.sciencedaily.com/releases/2024/10/241009183553.htm}}

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 to an idempotent element (possibly to 0). If the idempotent is non-zero,

it 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 {{harvs|last=Brauer|year1=1944|year2=1956|year3=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 {{harvs|last=Brauer|year1=1944|year2=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 {{harv|Brauer|1964|loc=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

{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=On the arithmetic in a group ring | jstor=87919 | mr=0010547 | year=1944 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=30 | issue=5 | pages=109–114 | doi=10.1073/pnas.30.5.109| pmid=16578120 | pmc=1078679| doi-access=free | bibcode=1944PNAS...30..109B }}
{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=On blocks of characters of groups of finite order I | jstor=87578 | mr=0016418 | year=1946 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=32 | issue=6 | pages=182–186 | doi=10.1073/pnas.32.6.182| pmc=1078910 | pmid=16578199| doi-access=free | bibcode=1946PNAS...32..182B }}
{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=On blocks of characters of groups of finite order. II | jstor=87838 | mr=0017280 | year=1946 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=32 | issue=8 | pages=215–219 | doi=10.1073/pnas.32.8.215| pmc=1078924 | pmid=16578207| doi-access=free | bibcode=1946PNAS...32..215B }}
{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=Zur Darstellungstheorie der Gruppen endlicher Ordnung | doi=10.1007/BF01187950 | mr=0075953 | year=1956 | journal=Mathematische Zeitschrift | issn=0025-5874 | volume=63 | pages=406–444| s2cid=186223039 }}
{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=Zur Darstellungstheorie der Gruppen endlicher Ordnung. II | doi=10.1007/BF01162934 | mr=0108542 | year=1959 | journal=Mathematische Zeitschrift | issn=0025-5874 | volume=72 | pages=25–46| s2cid=122127627 }}
{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=Some applications of the theory of blocks of characters of finite groups. I | doi=10.1016/0021-8693(64)90031-6 | mr=0168662 | year=1964 | journal=Journal of Algebra | issn=0021-8693 | volume=1 | issue=2 | pages=152–167| doi-access= }}
{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=On the first main theorem on blocks of characters of finite groups. | url=http://projecteuclid.org/euclid.ijm/1256053174 | mr=0267010 | year=1970 | journal=Illinois Journal of Mathematics | issn=0019-2082 | volume=14 | issue=2 | pages=183–187| doi=10.1215/ijm/1256053174 | doi-access=free }}
{{Citation | last1=Dade | first1=Everett C. | author1-link=Everett C. Dade | editor1-last=Powell | editor1-first=M. B. | editor2-last=Higman | editor2-first=Graham | editor2-link=Graham Higman | title=Finite simple groups. Proceedings of an Instructional Conference organized by the London Mathematical Society (a NATO Advanced Study Institute), Oxford, September 1969. | publisher=Academic Press | location=Boston, MA | isbn=978-0-12-563850-0 | mr=0360785 | year=1971 | chapter=Character theory pertaining to finite simple groups | pages=249–327}} gives a detailed proof of the Brauer's main theorems.
{{eom|id=b/b120440|first=H.|last= Ellers|title=Brauer's first main theorem}}
{{eom|id=b/b120450|first=H.|last= Ellers|title=Brauer height-zero conjecture}}
{{eom|id=b/b120460|first=H.|last= Ellers|title=Brauer's second main theorem}}
{{eom|id=b/b120470|first=H.|last= Ellers|title=Brauer's third main theorem}}
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}}

Category:Representation theory of finite groups

Category:Theorems in representation theory