康威-威尔士格与鲁德瓦利斯群及蒂茨群
The Conway-Wales lattice and the Rudvalis and Tits groups
- Kansas State University（堪萨斯州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文从交织汉明码ℋ₇与ℋ₈的Z₄自对偶偶码构造康威-威尔士格，确定其自同构群，导出鲁德瓦利斯群与蒂茨群，相关证明均在格内完成。
AI中文摘要：
我们从一个Z₄上的自对偶偶码C构造康威-威尔士格，其二进制层交织汉明码ℋ₇和ℋ₈。显式带符号对称性提升二进制剩余的每个自同构，完整带符号码群是高斯格的坐标框架稳定子。嵌入的缩放自对偶子格中的整反射提供额外对称性，由此恢复经典框架交换。计数固有定义的框架确定完整高斯等距群，其阶为583704576000。该群模去四个高斯标量后为鲁德瓦利斯单群，交叉稳定子的导出子群为阶17971200的蒂茨单群。构造、自同构群确定及两个单性证明均在格内完成，未假设已知群阶或使用识别或分类定理。
英文摘要：
We construct the Conway-Wales lattice from a self-dual even code $C$ over $\mathbb{Z}_4$ whose binary layers interweave the Hamming codes $\mathcal{H}_7$ and $\mathcal{H}_8$. Explicit signed symmetries lift every automorphism of the binary residue, and the full signed code group is the coordinate-frame stabilizer of the Gaussian lattice. An integral reflection in an embedded scaled self-dual sublattice supplies an additional symmetry, from which the classical frame exchange is recovered. Counting intrinsically defined frames determines the full Gaussian isometry group, of order $583\,704\,576\,000$. Its quotient by the four Gaussian scalars is the Rudvalis simple group, and the derived subgroup of a cross stabilizer is the Tits simple group of order $17\,971\,200$. The construction, automorphism-group determination, and both simplicity proofs are carried out within the lattice, without assuming the known group orders or using a recognition or classification theorem.
