Abstract
Let \([n] = K_1\dot{\cup }K_2 \dot{\cup }\cdots \dot{\cup }K_r\) be a partition of \([n] = \{1,2,\ldots ,n\}\) and set \(\ell _i = |K_i|\) for \(1\le i\le r\). Then the tuple \(P = \{K_1,K_2,\ldots ,K_r\}\) is an unordered partition of \([n]\) of shape \([\ell _1,\ldots ,\ell _r]\). Let \({{\mathcal {P}}}\) be the set of all partitions of \([n]\) of shape \([\ell _1,\ldots ,\ell _r]\). Given a fixed shape \([\ell _1,\ldots ,\ell _r]\), we determine all subgroups \(G\le S_n\) that are transitive on \({{\mathcal {P}}}\) in the following sense: Whenever \(P = \{K_1,\ldots ,K_r\}\) and \(P' = \{K_1',\ldots ,K_r'\}\) are partitions of \([n]\) of shape \([\ell _1,\ldots ,\ell _r]\), there exists \(g\in G\) such that \(g(P) = P'\), that is, \(\{g(K_1),\ldots ,g(K_r)\} = \{K_1',\ldots ,K_r'\}\). Moreover, for an ordered shape, we determine all subgroups of \(S_n\) that are transitive on the set of all ordered partitions of the given shape. That is, with \(P\) and \(P'\) as above, \(g(K_i) = K_i'\) for \(1\le i\le r\). As an application, we determine which Johnson graphs are Cayley graphs.
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The authors are indebted to the referees, whose suggestions greatly shortened the paper and improved its elucidation.
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Dobson, E., Malnič, A. Groups that are transitive on all partitions of a given shape. J Algebr Comb 42, 605–617 (2015). https://doi.org/10.1007/s10801-015-0593-2
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DOI: https://doi.org/10.1007/s10801-015-0593-2