具有有界非交换性的群的阿贝尔覆盖的Sharp渐近行为
Sharp Asymptotics for Abelian Covers of Groups with Bounded Noncommutativity
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中文总结 AI 辅助
该研究确定了具有有界非交换性的群的阿贝尔覆盖最小数目的精确指数增长率，证明了相关定量估计，解决了厄尔多斯问题#117。
中文摘要 AI 辅助
我们确定了覆盖具有有界两两非交换性的群所需的阿贝尔子群的最小数目的精确指数增长率。设ω(G)表示群G的两两非交换子集的最大规模，a(G)为阿贝尔覆盖的最小规模，定义h(n)=sup{a(G):ω(G)≤n}。我们证明了定量估计log₂h(n)=n/2+O(√n(log(n+2))³)，因此h(n)^(1/n)→√2。特殊2-群给出了匹配的下界。对于上界，我们通过同余归约到有限群，通过导出子群的中心列和交替换位子形式分析有限p-群，控制中心因子间的相互作用，然后通过西罗分解和幂零子群（Fitting子群）以多项式代价传递。该论证还确定了阿贝尔子群最小指数的相同精确指数率，并表明渐近极值集中在2-群中。该结果在精确指数渐近水平上解决了厄尔多斯问题#117。
英文摘要
We determine the sharp exponential growth rate of the minimum number of abelian subgroups required to cover a group with bounded pairwise noncommutativity. Let $ω(G)$ denote the largest size of a pairwise noncommuting subset of a group $G$, let $a(G)$ be the least size of an abelian cover, and define $h(n)=\sup\{a(G):ω(G)\le n\}$. We prove the quantitative estimate $\log_2 h(n)=n/2+O(\sqrt{n}\,(\log(n+2))^3)$, and hence $h(n)^{1/n}\to\sqrt{2}$. Extraspecial $2$-groups give the matching lower bound. For the upper bound, we reduce to finite groups by isoclinism, analyze finite $p$-groups through a central series of the derived subgroup and alternating commutator forms, control interactions between central factors, and then pass through Sylow decomposition and the Fitting subgroup at polynomial cost. The argument also determines the same sharp exponential rate for the least possible index of an abelian subgroup and shows that asymptotic extremality is concentrated in $2$-groups. This result resolves Erdős Problem #117 at the level of its sharp exponential asymptotics.
