Characters of p′-Degree of p-Solvable Groups-猫眼课题宝

p-可解群的p′-度的性质

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2001

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作者：

I. Isaacs;G. Navarro

确定有限群G的素数p和Sylow p-子群P，记N=NG P。(仍未被证明的)“McKay猜想”断言Irrp‘G=Irrp’N，或者换句话说，G和N具有相同数目的不可约特征标，其次数不能被p整除。(J.McKay在1971年提出了他的猜想，但仅对简单群且仅对p=2；Alperin于1975年在[1]中给出了该猜想的一般形式的第一个形式陈述。)麦凯猜想现在已知适用于许多类别的群。1973年，第一作者证明了奇数阶群和p=2时的所有可解群的猜想(这一结果见[3])。随后，T.R.Wolf给出了一个对每个可解群都有效的证明，E.C.Dade对所有p-可解群证明了这个猜想。Okuyama和Wajima在[10]中给出了p-可解群的一个更简单的证明。(由于我们在本文中只关注p-可解群，所以我们不会明确地提到许多其他已知的情况。)

Fix a prime p and a Sylow p-subgroup P of a finite group G, and write N = NG P . The (still unproved) “McKay conjecture” asserts that Irrp′ G = Irrp′ N , or in other words that G and N have equal numbers of irreducible characters with degrees not divisible by p. (J. McKay proposed his conjecture in 1971, but only for simple groups and only for p = 2; the first formal statement of the general form of the conjecture was given in 1975 by Alperin in [1].) The McKay conjecture is now known to be valid for many classes of groups. In 1973, the first author proved the conjecture for (solvable) groups of odd order and for all solvable groups when p = 2. (This result appears [3].) Subsequently, T. R. Wolf gave a proof valid for every solvable group and E. C. Dade verified the conjecture for all p-solvable groups. A simpler proof for p-solvable groups was then given by Okuyama and Wajima in [10]. (Since our concern in this paper is only with p-solvable groups, we will not mention explicitly the many other known cases.)

Characters of p′-Degree of p-Solvable Groups

p-可解群的p′-度的性质

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