Fields Generated by Characters of Finite Groups
Fields Generated by Characters of Finite Groups
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由有限群的特征生成的域
DOI：
10.1112/jlms/s2-4.4.735
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发表时间：
1972
影响因子：
1.2
通讯作者：
B. Gordon
中科院分区：
文献类型：
--
作者：
B. Fein;B. Gordon
本文研究有限群G的复特征标表的元素在有理数Q上生成的域#（G）的性质。我们刻画了#（G）是Q阿贝尔扩张，且具有如下性质：每个有限剩余类域是G的极小分裂域。虽然一般来说#（G）本身不是G的分裂域，但G的每一个特征为零的分裂域都包含#（G）的同构副本作为子域。作为度量G的Schur指数m（G）与G的分裂域有多接近的度量，我们定义G的Schur指数m（G）为最小值[K：G）]，最小值取于G的所有分裂域K上，使得K 2 #（G）.我们证明了m（G）具有表示的Schur指标所具有的许多性质，特别地，我们证明了m（G）整除G的任意Abel正规子群的指标.最后，我们考虑的问题，哪些领域可以产生的邻接条目的字符表。证明了Q的每一个阿贝尔扩张都有一个本原元是某个有限群的特征标表的一个条目。对于有限群的特征标表的一行或一列在Q上生成的域，我们也得到了类似的结果。在本文中，G将表示阶的有限群|G|.我们用^（K）表示代数数域K的整数环。设P是J（K）的素理想，我们设（p）= PnZ，pa.有理数素数和Z是Q中的整数环。我们用KP表示K的P-adic完备化，用KP表示KP的有限剩余类域。对于aeJ（K），我们用a表示K在P（它与KP同构）的剩余类域的元素a+ P。我们将a视为KP的一个元素。所谓G的K-表示，我们指的是G由元素在K中的矩阵表示。A. k字符将是a提供的字符。^-表示. en将表示Q上的单位的本原n次根，并且我们设Qn= Q（en）。Zp表示特征为p的素域，p^ 0，Zp是Zp的代数闭包。P（G）表示在Zp上生成的域，其值%（g）取在所有G上和G的所有不可约Zp-特征标%上。o（G）=#（G）。对于任何域L，我们用L [G]表示G在L上的群代数。如果L [G]/rad（L [G]）是L上完备矩阵代数的直和，则L是G的分裂域.如果L和K是域，且L是K的有限正规扩张，则我们用&（L\K）表示L在K上的伽罗瓦群。我们建议读者参考[3]、[7]和[1]，以了解相关的表示论、代数数论和假设的代数理论。
In this paper we investigate the properties of the field#(G) generated over the rationals, Q, by the entries of the complex character table of a finite group G. We characterize#(G) as that Abelian extension of Q with the property that every finite residue class field is a minimal splitting field for G. Although#(G) is not, in general, itself a splitting field for G, every splitting field of characteristic zero for G contains an isomorphic copy of#(G) as a subfield. As a measure of how close#(G) is to being a splitting field for G, we define the Schur index, m (G), of G to be minimum [K:#(G)], the minimum being taken over all splitting fields K for G, such that K 2#(G). We show that m (G) has many of the properties possessed by the Schur index of a representation, and, in particular, we show that m (G) divides the index of any Abelian normal subgroup of G. Finally we consider the question of which fields can be generated by adjoining entries of character tables. We prove that every Abelian extension of Q has a primitive element which is an entry of the character table of some finite group. We also obtain the analogous results for the fields generated over Q by one row or one column of the character table of a finite group. Throughout this paper G will denote a finite group of order| G|. We denote the ring of integers of an algebraic number field K by^(K). For P a prime ideal of J {K), we put (p)= P nZ, p a. rational prime and Z the ring of integers in Q. We denote the P-adic completion of K by KP and the finite residue class field of KP by KP. For aeJ (K), we denote the element a+ P of the residue class field of K at P (which is isomorphic to KP) by a. We view a as an element of KP. By a K-representation of G we shall mean a representation of G by matrices with entries in K. A. K-character will be the character afforded by a.^-representation. en will denote a primitive nth root of unity over Q and we put Qn= Q (en). Zp will denote the prime field of characteristic p, p^ 0, and Zp will be an algebraic closure of Zp.# P (G) will denote the field generated over Zp by the values%{g) taken over allg e G and over all irreducible Zp-characters% of G.^ o (G)=#(G). For any field L we denote the group algebra of G over L by L [G]. L is a splitting field for G if L [G]/rad (L [G]) is a direct sum of complete matrix algebras over L. If L and K are fields with L a finite normal extension of K, we denote the Galois group of L over K by & (L\K). We refer the reader to [3],[7], and [1] for the relevant representation theory, algebraic number theory, and theory of algebras assumed.