Enumerating p-Groups, II: Problems Whose Solution is PORC
Enumerating p-Groups, II: Problems Whose Solution is PORC
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DOI：
10.1112/plms/s3-10.1.566
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发表时间：
1960
影响因子：
1.8
通讯作者：
G. Higman
中科院分区：
文献类型：
--
作者：
G. Higman
1.1.导论. PORC函数A函数f（x），其定义域D和值域是复数的集合，可以说是E上的多项式，其中E是复数的集合，如果存在多项式g（x）使得f（x）= g（x）对于D和E的交集中的所有x。在本文中，我们感兴趣的功能，其领域是集的整数，并在问题是否有一个整数n这样的功能是多项式对每个剩余类modfi，在这个意义上。对于函数f（x）到lf（x）存在这样一个整数，我们将证明这个命题是PORC 1。正如在（4）的引言中所解释的，现有的证据表明，对于固定的n，作为素数p的函数，pn阶群的同构类的数目可能是PORC。正是研究这种可能性的愿望激发了目前的讨论，尽管我们得到的结果离解决这个问题还有很长的路要走。我们的主要定理描述了一类计数问题，涉及q元有限域，q是素数幂，其解作为q的函数是PORC，这些定理在§ 1.2中叙述，这里我们引用一些有代表性的例子。
1.1. Introduction. PORC functions Afunction f {x), whose domain D and range are sets of complex numbers, may be said to be polynomial on E, where E is a set of complex numbers, if there is a polynomial g (x) such that f (x)= g {x) for all x in the intersection of D and E. In this paper we are interested in functions whose domains are sets of integers, and in the question whether there is an integer n such that the function is polynomial on each residue class modfi, in this sense. We shall abbreviate the statement that such an integer exists for the function f (x) to lf (x) is PORC1. As was explained in the introduction to (4), the available evidence suggests that for fixed n the number of isomorphism classes of groups of order pn, considered as a function of the prime p, may be PORC. It was the desire to investigate this possibility that motivated the present discussion, though the results that we obtain fall a long way short of settling the question. Our main theorems describe classes of enumeration problems involving the finite field of q elements, q a prime power, whose solutions considered as functions of q are PORC. These theorems are stated in § 1.2, here we quote some representative cases.