McKay-Alperin conjecture

Let $G$ be a finite group. For any prime number $p$, let $m_p(G)$ be the number of irreducible complex characters of $G$ with degree prime to $p$ (cf. also Character of a group). The simplest form of the McKay–Alperin conjectures asserts that

$$m_p(G)=m_p(N_G(P)),$$

where $P$ is a Sylow $p$-subgroup of $G$ and $N_G(P)$ is its normalizer (cf. also Sylow subgroup; $p$-group; Normalizer of a subset). J. McKay [a2] first suggested this might be true when $G$ is a simple group. J.L. Alperin [a1] observed that it is probably true for all finite groups.

Alperin also made a more general conjecture, involving characters in $p$-blocks. (See Brauer first main theorem for notation and definitions.) Let $B$ be a $p$-block of $G$ with defect group $D$ (cf. Defect group of a block), and let $\chi$ be an irreducible character (cf. also Irreducible representation) belonging to $B$. Let $P$ be a Sylow $p$-subgroup of $G$. By a theorem of R. Brauer, $|P|/|D|$ divides the degree $\chi(1)$. The character $\chi$ is said to have height zero if the largest power of $p$ dividing $\chi(1)$ is $|P|/|D|$. The more general Alperin conjecture asserts that the number of irreducible characters of height zero in $B$ is equal to the number of irreducible characters of height zero in the unique block of $N_G(D)$ sent to $B$ by the Brauer correspondence.

The conjectures are still not proved (1998), but the evidence in their favour is very strong.

References

[a1]	J.L. Alperin, "The main problem of block theory" W.R. Scott (ed.) F. Gross (ed.) , Proc. Conf. Finite Groups (Park City, Utah, 1975) , Acad. Press (1976)
[a2]	J. McKay, "Irreducible representations of odd degree" J. Algebra , 20 (1972) pp. 416–418