On a projective representation of the Hall-Janko group
On a projective representation of the Hall-Janko group
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关于 Hall-Janko 群的射影表示
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发表时间：
1968
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影响因子：
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通讯作者：
Walter Feit
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作者：
Walter Feit
在对具有复系数的六阶射影群的研究中，我们发现，Marshall Hall 和 David Wales 在一篇论文中描述的 Hall-Janko 群（唯一的 604,800 阶简单群）具有六阶射影表示。事实上，存在一个唯一群 &，其中心为 Z，二阶，且 &/Z £ËG，满足以下条件： (A) G 的 Sylow-7 子群由四阶元素归一化； (B) 6048阶G的子群J7s(3)在同态Gi~*Gi/Z SG下的逆像与ZX Uz(3)同构。 Gi 在复数域中有两个共轭、忠实、不可约的六阶表示。该表示可以写成 Q(/5, V ~ 7 ) 。 G的字符表是可以给定的，并且是唯一的。 G 的忠实、不可约表示的度数为 6、6、64、64、50、50、216、14、84、126、126、252、56、56、448、350、336。G 的存在性通过采用 (mod 3) 六度表示来验证。这种仅限于 Z Uz(3) 的模块化表示具有三维不变子空间 V。然后，在 Gi 下，V 有一百个图像，G 的生成器对这些图像进行排列，就像它们的图像对 Hall-Wales 论文中描述的字母进行排列一样。这是通过计算机使用乔治·夏皮罗编写的程序进行检查的。在将 Sylow-7 子群的归一化器写成正规形式后，获得了复数域上唯一的酉矩阵，用于 Gi 生成元的六维表示。
In the line of an investigation of the projective groups of degree six with complex coefficients, it was discovered that the Hall-Janko group, the unique simple group of order 604,800, described in a paper by Marshall Hall and David Wales, has a projective representation of degree six. In fact, there exists a unique group, &, with center, Z, of order two, and &/Z £ËG, satisfying the following: (A) the Sylow-7-subgroup of G is normalized by an element of order four; (B) the inverse image, under the homomorphism Gi~*Gi/Z SG, of a subgroup, J7s(3), of G of order 6048 is isomorphic to ZX Uz(3). Gi has two conjugate, faithful, irreducible representations of degree six in the complex field. This representation can be written in Q(/5, V ~ 7 ) . The character table of G can be given, and it is unique. The degrees of the faithful, irreducible representations of G are 6, 6, 64, 64, 50, 50, 216, 14, 84, 126, 126, 252, 56, 56, 448, 350, 336. Existence of G was verified by taking (mod 3) a representation of degree six. This modular representation restricted to Z Uz(3) has a three dimensional invariant subspace, V. Then, under Gi, V has one hundred images, which generators of G permute exactly as their images permute the letters described in the Hall-Wales paper. This was checked by computer with a program written by George Shapiro. Unique unitary matrices over the complex field were obtained for a six dimensional representation of generators of Gi after the normalizer of a Sylow-7-subgroup was written in a normal form.