# Difference between revisions of "McKay-Alperin conjecture"

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− | Let | + | {{TEX|done}} |

+ | Let $G$ be a [[Finite group|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|Character of a group]]). The simplest form of the McKay–Alperin conjectures asserts that | ||

− | + | $$m_p(G)=m_p(N_G(P)),$$ | |

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

− | Alperin also made a more general conjecture, involving characters in | + | Alperin also made a more general conjecture, involving characters in $p$-blocks. (See [[Brauer first main theorem|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|Defect group of a block]]), and let $\chi$ be an irreducible character (cf. also [[Irreducible representation|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. | The conjectures are still not proved (1998), but the evidence in their favour is very strong. | ||

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====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. McKay, "Irreducible representations of odd degree" ''J. Algebra'' , '''20''' (1972) pp. 416–418</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> 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)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J. McKay, "Irreducible representations of odd degree" ''J. Algebra'' , '''20''' (1972) pp. 416–418</TD></TR></table> | ||

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+ | [[Category:Group theory and generalizations]] |

## Latest revision as of 18:45, 13 October 2014

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 |

**How to Cite This Entry:**

McKay-Alperin conjecture.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=McKay-Alperin_conjecture&oldid=13806