AI 中文总结
研究\(p\)-进域中可定义群的性质，证明其弱正则性，表明单参数子群等可定义，得出\(G_u\)与\(G_\Omega\)的可定义性及\(G_u\)与\(H\)相关结论，尤其在可定义顺从时\(G_u = H_u\)。
AI 中文摘要
我们证明了在纯\(p\)-进域\(\mathbb{Q}_p\)中可定义的每个群\(G\)都是弱正则的。表明每个单参数子群是可定义的，其单参数核\(G_u\)与合适的正则开子群\(G_\Omega\)是可定义的。还表明若\(H\)是\(G\)的一个\(dfg\)分量，则\(G_u=\langle(H^g)_u:g\in G\rangle\)。特别地，当\(G\)是可定义顺从的时，\(G_u = H_u\)。
英文摘要
We prove that every group $G$ definable in the pure $p$-adic field $\mathbb{Q}_p$ is weakly regular. We show that every one-parameter subgroup is definable. We also show that its one-parameter core $G_u$, together with a suitable regular open subgroup $G_Ω$, is definable. Finally, we show that if $H$ is a dfg component of $G$, then $G_u=\langle(H^g)_u:g\in G\rangle$. In fact, $G_u$ is a finite product of the one-parameter cores of finitely many conjugates of $H$. In particular, when $G$ is definably amenable, $G_u=H_u$.
