Isaacs-Navarro conjecture

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The Isaacs-Navarro conjecture is a slight generalization of the McKay conjecture and is believed to hold for all finite groups. The conjecture was introduced in a 2002 paper by Isaacs and Navarro.

Statement

Suppose G is a finite group and p is a prime number. Denote by f(G,p,a) the number of equivalence classes of irreducible representations of G over the complex numbers whose degree is congruent to a or -a modulo p. Then, if a is not divisible by p, and P is a p-Sylow subgroup of G, we have:

\! f(G,p,a) = f(N_G(P),p,a)

Current status

The conjecture for all finite groups is open, but it has been resolved for some types of groups.

Group property or group family Status of the conjecture Explanation
finite nilpotent group resolved obvious: the normalizer of any Sylow subgroup is the whole group.
symmetric group on finite set resolved See Fong's 2003 paper on the conjecture.
alternating group on finite set resolved See Nath's 2009 paper on the conjecture.
double cover of symmetric group and double cover of alternating group resolved in odd characteristic See Gramain's 2011 paper on the conjecture.

References