Isaacs-Navarro conjecture
From Groupprops
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 is a finite group and is a prime number. Denote by the number of equivalence classes of irreducible representations of over the complex numbers whose degree is congruent to or modulo . Then, if is not divisible by , and is a -Sylow subgroup of , we have:
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
- New refinements of the McKay conjecture for finite groups by I. Martin Isaacs and Gabriel Navarro, Annals of Mathematics, Volume 156, Page 333 - 344(Year 2002): ^{ArXiV copy}^{More info}
- The Isaacs–Navarro conjecture for symmetric groups by Paul Fong, Journal of Algebra, ISSN 00218693, Volume 260,Number 1, Page 154 - 161(February 2003): ^{Official copy (gated)}^{More info}
- The Isaacs–Navarro conjecture for the alternating groups by Rishi Nath, Journal of Algebra, ISSN 00218693, Volume 321,Number 6, Page 1632 - 1642(March 2009): ^{Official copy (gated)}^{More info}
- The Isaacs–Navarro conjecture for covering groups of the symmetric and alternating groups in odd characteristic by Jean-Baptiste Gramain, Journal of Algebraic Combinatorics, Volume 34,Number 3, Page 401 - 426(November 2011): ^{Official copy (gated)}^{More info}