无立方因子阶的群的数量
The number of groups of cubefree order
- Monash University（蒙纳士大学）
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AI总结：
该研究推广Hölder的无平方因子阶群计数公式，给出无立方因子阶群同构类数量的精确组合公式，推导其渐近结果并改进相关猜想的界，指数2最优且强形式对几乎所有无立方因子阶成立
AI中文摘要：
推广Hölder在1895年提出的无平方因子阶群的经典计数方法，我们给出了给定无立方因子阶的群同构类数量的精确公式。历经130多年后，这是首个覆盖范围远大于无平方因子阶的此类公式（覆盖所有整数的83%，而无平方因子阶仅覆盖61%）。与Hölder的公式类似，我们的公式是组合性的：可通过阶的素因数分解，经算术运算和查表得出结果，无需构造任何一个群。该公式的结构可导出无立方因子群自然子类的计数公式，应用于计算群论领域。我们还推导了新的渐近结果。Blackburn等人（2007）猜想，无立方因子阶n的群数量gnu(n)满足gnu(n)<n²。我们证明gnu(n)≤n^{2+o(1)}，改进了他们综述中记载的gnu(n)<n⁸的界，且指数2是最优的，即存在无穷多个无立方因子阶n使得gnu(n)≥n^{2-o(1)}。最后，我们证明该猜想的一个强形式对几乎所有无立方因子阶成立，即gnu(n)≤(log n)^{(log log n)^{O(1)}}。
英文摘要：
Generalising Hölder's classical group enumeration for squarefree orders (1895), we provide an exact formula for the number of isomorphism types of groups of a given cubefree order. After more than 130 years, this is the first such formula that covers significantly more orders than the squarefree ones (83% versus 61% of all integers). Like Hölder's formula, ours is combinatorial: it can be evaluated from the prime factorisation of the order by arithmetic operations and table look-ups, without constructing a single group. The structure of our formula leads to counting formulas for natural subclasses of cubefree groups, with applications in computational group theory. We also derive new asymptotic results. Blackburn et al. (2007) conjectured that the number gnu(n) of groups of cubefree order n satisfies gnu(n)<n^2. We show that gnu(n)\leq n^{2+o(1)}, which improves the bound gnu(n)<n^8 recorded in their survey, and we prove that the exponent 2 is best possible, that is, gnu(n)\geq n^{2-o(1)} for infinitely many cubefree n. Lastly, we show that a much stronger form of the conjecture holds for almost every cubefree order, namely, \gnu(n)\leq (\log n)^{(\log\log n)^{O(1)}}.
