The Number of Irreducible Character Degrees of Solvable Groups Satisfying the One-Prime Hypothesis
The Number of Irreducible Character Degrees of Solvable Groups Satisfying the One-Prime Hypothesis
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DOI：
10.1007/s10468-005-3596-1
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发表时间：
2005-10
影响因子：
0.6
通讯作者：
M. Lewis
中科院分区：
文献类型：
--
作者：
M. Lewis
设G是一个有限群，记cd（G）为G的不可约特征标的次数集。我们说G满足单素假设，如果当a和在cd（G）中具有不同的次数时，则a和的最大公约数是1或素数。我们证明了如果G是满足单素假设的可解群，则|cd（G）|≤9。我们还构造了满足单素假设的可解群G，|cd（G）|=9，这表明本文中找到的界是最佳可能界。
LetGbe a finite group, and write cd(G) for the set of degrees of irreducible characters ofG. We sayGsatisfies the one-prime hypothesis if wheneveraandbare distinct degrees in cd(G), then the greatest common divisor ofaandbis either 1 or a prime. We show that ifGis a solvable group satisfying the one-prime hypothesis, then |cd(G)|≤9. We also construct a solvable groupGsatisfying the one-prime hypothesis with |cd(G)|=9 which shows that the bound found in this paper is the best possible bound.