In mathematics, specifically in the field of

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, the McKay conjecture is a

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

of equality between the number of irreducible complex

characters Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...

of degree not divisible by a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

p

to that of the

normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', ...

of a Sylow

p

-subgroup. It is named after Canadian mathematician John McKay.

Statement

Suppose

p

is a prime number,

G

is a finite group, and

P \leq G

is a Sylow

p

-subgroup. Define :

\textrm_(G) := \

where

\textrm(G)

denotes the set of complex irreducible characters of the group

G

. The McKay conjecture claims the equality :

, \textrm_(G),  = , \textrm_(N_G(P)),

where

N_G(P)

is the normalizer of

P

G

References

* (Corrected reprint of the 1976 original, published by Academic Press.) * {{refend Representation theory of groups