On the Number of Generators and Composition Length of Finite Linear Groups
On the Number of Generators and Composition Length of Finite Linear Groups
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关于有限线性群的生成元数和组合长度
DOI：
10.1006/jabr.2001.8882
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发表时间：
2001
影响因子：
0.9
通讯作者：
M. Morigi
中科院分区：
文献类型：
--
作者：
A. Lucchini;F. Menegazzo;M. Morigi
在1991年狄克逊和Kovacs 8表明，对于每个域K有有限的程度，其素子域有一个数字d，使每一个K有限幂零不可约线性群的程度n2的K可以'产生的d n log n元。之后，Bryant等人证明了K对于可解线性群也是如此，这导致了一个问题，即类似的结果是否也可以成立，也可以去除可解性假设。在15它证明了答案是积极的特殊情况下有限领域。在本文中，我们能够处理数域的情况下，从而给一个完整的解决方案的问题。也就是说，.用d G表示群G的生成元个数，我们证明：
In 1991 Dixon and Kovacs 8 showed that for each field K which has finite degree over its prime subfield there is a number d such that every K finite nilpotent irreducible linear group of degree n 2 over K can be ' generated by d n log n elements. Afterwards Bryant et al. 3 proved K that the same is true for solvable linear groups and this led to asking whether a similar result could hold also removing the solvability hypothesis. In 15 it is proved that the answer is positive in the particular case of finite fields. In the present paper we are able to deal with the case of number fields, thus giving a complete solution to the problem. Namely, Ž . denoting by d G the number of generators of a group G, we prove: