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Isometries between finite groups. (English) Zbl 1478.94149

Summary: 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 C. Carlet [IEEE Trans. Inf. Theory 44, 1543–1547 (1998; Zbl 0935.94028)] for \(G=\mathbb{Z}_{2^k}\) and B. Yildiz and Z. Ödemiş Özger [TWMS J. Appl. Eng. Math. 2, No. 2, 145–153 (2012; Zbl 1280.94113)] 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 \(d_{RT}\).

MSC:

94B65 Bounds on codes
94B05 Linear codes (general theory)
20C99 Representation theory of groups
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References:

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