用类函数细化有限群的不变量
Refining invariants of finite groups with class functions
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中文总结 AI 辅助
本文研究有限群中由类函数导出的数值不变量，证明其广义特征在两种重要情形下为特征，刻画子群相容性，并借助McKay猜想给出阿贝尔Sylow $p$-子群的判据。
中文摘要 AI 辅助
有限群的许多数值不变量都源于自然关联的类函数中平凡特征的重数，而其余重数则携带更精细的结构信息。对于其阶仅涉及选定素数集的元素的共轭类数目，自然的类函数是一个广义特征，它在经典共轭特征与正则特征之间进行插值。我们证明在两个重要的普遍情形下它是一个特征，这两个情形由Robinson独立获得，并且我们排除了一个族群和特征，我们认为在这些情形中最可能出现反例。此外，我们刻画了这些类函数何时与传递到子群相容，从而加强了Sangroniz的一个定理。对于实元素，类似的类函数被证明总是给出一个特征。最后，我们给出一个对偶构造，结合最近证明的McKay猜想，得到一个有限群具有阿贝尔Sylow $p$-子群的最佳可能判据。
英文摘要
Many numerical invariants of a finite group arise as the multiplicity of the trivial character in a naturally associated class function, and the remaining multiplicities carry finer structural information. For the number of conjugacy classes of elements whose order involves only a chosen set of primes, the natural such class function is a generalized character interpolating between the classical conjugating character and the regular character. We prove that it is a character in two important general cases, obtained independently by Robinson, and we rule out a family of groups and characters in which, we argue, a counterexample would be most likely to arise. Furthermore, we characterize when these class functions are compatible with passing to a subgroup and thereby sharpen a theorem of Sangroniz. The analogous class function for real elements is shown to always give a character. Finally, we give a dual construction that, together with the recently proved McKay Conjecture, yields a best-possible criterion for a finite group to have an abelian Sylow $p$-subgroup.
发表机构
- HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）
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