Given the ring of integers O K of an algebraic number field K, for which natural numbers n there exists a finite group G ⊂ GL(n, O K ) such that O K G, the O K -span of G, coincides with M(n, O K ), the ring of (n × n)-matrices over O K ? The answer is known if n is an odd prime. In this paper we study the case n = 2; in the cases when the answer is positive for n = 2, for n = 2m there is also a finite group G ⊂ GL(2m, O K ) such that O K G = M(2m, O K ).
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机译：给定一个代数数字段K的整数O K 的环，对于自然数n，存在一个有限群G⊂GL（n，O K ）使得O G的O K 跨度 K G与（n×n）的环M（n，O K ）重合-O K 上的矩阵？如果n是奇数素数，则答案是已知的。在本文中，我们研究了n = 2的情况；在n = 2的答案为肯定的情况下，对于n = 2m，还有一个有限群G⊂GL（2m，O K ）使得O K G = M（2m，O K ）。
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