论文标题
有限组之间的异构体
Isometries between finite groups
论文作者
论文摘要
我们证明，如果$ h $是任何环状组$ g $的索引$ n $的子组，则可以将$ g $在$（h^n，d _ {_ {ham}}^n）$中嵌入为$（h^n，d _ {_ {ham}}^n）$，从而将$ g = \ nathbb的先前结果（1998）推广为$ g = \ nathbb {z} _} $ and y mathbb {z} _} _ { $ g = \ mathbb {z} _ {p^k} $带有$ p $ prime。接下来，对于任何积极的整数$ q $，我们定义$ Q $ -ADIC $ d_q $ in $ \ MATHBB {z} _ {q^n} $，并证明$（\ Mathbb {z} _ {q^n}，d_q），d_q）$均为$（\ m m mathbbbb {$ nathbbbb {$ m马克） $ n $，其中$ d_ {rt} $是rosenbloom-tsfasman公制。然后，我们证明，对于某些可以明确构建的指标，相同基数的任何一对有限组都是等距的。最后，我们考虑了给定组的子组的链条$ \ MATHCAL {C} $，并定义了链公制$ d _ {\ Mathcal {c}} $和两个链之间的链异分析。让$ g，k $为$ | g | = q^n $，$ | k | = q $，让$ h <g $。使用链条，我们证明在某些条件下，$（g，d_ \ nathcal {c}）\ simeq（k^n，d_ {rt}）$和$（g，d_ \ d_ \ nathcal {c}）概括$ d_ {rt} $的度量。
We prove that if $H$ is a subgroup of index $n$ of any cyclic group $G$, then $G$ can be isometrically embedded in $(H^n, d_{_{Ham}}^n)$, thus generalizing previous results of Carlet (1998) for $G=\mathbb{Z}_{2^k}$ and Yildiz-Özger (2012) for $G=\mathbb{Z}_{p^k}$ with $p$ prime. Next, for any positive integer $q$ we define the $q$-adic metric $d_q$ in $\mathbb{Z}_{q^n}$ and prove that $(\mathbb{Z}_{q^n}, d_q)$ is isometric to $(\mathbb{Z}_q^n, d_{RT})$ for every $n$, where $d_{RT}$ is the Rosenbloom-Tsfasman metric. More generally, we then demonstrate that any pair of finite groups of the same cardinality are isometric to each other for some metrics that can be explicitly constructed. Finally, we consider a chain $\mathcal{C}$ of subgroups of a given group and define the chain metric $d_{\mathcal{C}}$ and chain isometries between two chains. Let $G, K$ be groups with $|G|=q^n$, $|K|=q$ and let $H<G$. Using chains, we prove that under certain conditions, $(G,d_\mathcal{C}) \simeq (K^n, d_{RT})$ and $(G,d_\mathcal{C}) \simeq (H^{[G:H]}, d_{BRT})$ where $d_{BRT}$ is the block Rosenbloom-Tsfasman metric which generalizes $d_{RT}$.
