AI 中文总结
该研究定义Burnside环单位的正规核压缩映射，推导其系数性质，给出核平凡单位的刻画，证明有限群中核平凡单位群的秩无界。
AI 中文摘要
设$G$为有限群，我们研究$G$的Burnside环上的加法映射，该映射将轨道$[G/H]$发送到$[G/\operatorname{core}_G(H)]$，我们称之为正规核压缩，并探究其在单位上的行为。对于正规子群$N\trianglelefteq G$，我们证明累积$N$-核系数等于$G/N$上$N$-不动点元素的轨道基系数之和。因此，对于单位而言，该系数仅取$-1,0,1$，而单个核系数可通过正规子群格上的莫比乌斯反演得到。利用Yoshida准则，我们进一步将累积核系数表示为$N$-标记和$G/N$的线性特征的函数，这给出了正规部分Burnside环中存在非零$N$-核系数的单位的充要条件：$G/N$必须是初等阿贝尔2-群。另一方面，被核压缩映射到$1_{\Omega(G)}$的单位（称为核平凡单位）满足所有核循环子群处的标记均等于1的条件。利用有理Burnside环的本原幂等元与Yoshida准则，我们将核平凡单位群实现为${\mathbb F}_2$-线性缺陷映射的核。最后，我们得到该缺陷映射沿正规商的分裂及沿直积的直和分解，结合直积分解与$S_4$的非平凡示例，证明有限群中核平凡单位群的秩是无界的。
英文摘要
Let $G$ be a finite group. We study the additive map on the Burnside ring of $G$ that sends an orbit $[G/H]$ to $[G/\operatorname{core}_G(H)]$, which we call the \emph{normal-core compression}, and investigate its behavior on units. For a normal subgroup $N\trianglelefteq G$, we show that the cumulative $N$-core coefficient is equal to the sum of the orbit-basis coefficients of the $N$-fixed-point element over $G/N$. Consequently, for a unit this coefficient takes only the values $-1,0,1$, while the individual core coefficients are recovered by Möbius inversion on the lattice of normal subgroups. Using Yoshida's criterion, we further express the cumulative core coefficient in terms of the $N$-mark and a linear character of $G/N$. This yields a necessary and sufficient condition for the existence of a unit with nonzero $N$-core coefficient in the normal partial Burnside ring: $G/N$ must be an elementary abelian $2$-group. On the other hand, the units mapped to $1_{Ω(G)}$ by the core compression, called \emph{core-trivial units}, are characterized by the condition that all marks at core-cyclic subgroups are equal to $1$. Using primitive idempotents of the rational Burnside ring and Yoshida's criterion, we realize the group of core-trivial units as the kernel of an ${\mathbb F}_2$-linear defect map. Finally, we obtain a splitting along normal quotients and a direct-sum decomposition along direct products for this defect map. Combining the direct-product decomposition with an explicit nontrivial example for $S_4$, we show that the ranks of core-trivial unit groups are unbounded among finite groups.
Comments32 pages
