A reduction theorem for the McKay conjecture
A reduction theorem for the McKay conjecture
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DOI：
10.1007/s00222-007-0057-y
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发表时间：
2007-05
影响因子：
3.1
通讯作者：
I. Isaacs;G. Malle;G. Navarro
中科院分区：
文献类型：
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作者：
I. Isaacs;G. Malle;G. Navarro
麦凯猜想断言，对于每个有限群 G 和每个素数，Ghavingp' 度的不可约字符数等于 G 的 Sylowp 子群的归一化子的此类字符数。尽管这已在大量群（包括所有可解群和所有对称群）中得到证实，但尚未找到一般证明。在本文中，我们将麦凯猜想简化为有关简单群的问题。我们给出了我们希望所有简单群都满足的条件列表，并且我们表明，如果涉及 G 的每个简单群都满足这些条件，则麦凯猜想对于有限群 G 成立。此外，我们还确定对于所有素数幂 q≥4 的简单群 PSL2(q) 以及 Suzuki 群 Sz(q) 和 Ree 群 R(q) 都满足我们的条件，其中 q=2e 或 q=3，并且 e>1 是奇数。由于零星单群 J1 也满足我们的条件，因此麦凯猜想对于具有阿贝尔 Sylow 2 子群的每个有限群都成立（对于所有素数 sp）。
The McKay conjecture asserts that for every finite groupGand every primep, the number of irreducible characters ofGhavingp’-degree is equal to the number of such characters of the normalizer of a Sylowp-subgroup ofG. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite groupGif every simple group involved inGsatisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL2(q) for all prime powersq≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), whereq=2eorq=3erespectively, ande>1 is odd. Since our conditions are also satisfied by the sporadic simple groupJ1, it follows that the McKay conjecture holds (for all primesp) for every finite group having an abelian Sylow 2-subgroup.