Normal-Core Compression for Units in Burnside Rings

Abstract

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 M\"obius 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 $1_{\Omega(G)}$ 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.