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Wreath product of permutation groups and their actions on a sets. (English) Zbl 1499.20074

Summary: The object of wreath product of permutation groups is defined the actions on cartesian product of two sets. In this paper we consider \(S(\Gamma)\) and \(S(\Delta)\) be permutation groups on \(\Gamma\) and \(\Delta\) respectively, and \(S(\Gamma)^\Delta\) be the set of all maps of \(\Delta\) into the permutations group \(S(\Gamma)\). That is \(S(\Gamma)^\Delta=\{f:\Delta \rightarrow S(\Gamma)\}\). \(S(\Gamma)^\Delta\) is a group with respect to the multiplication defined by for all \(\delta\) in \(\Delta\) by \((f_1f_2)(\delta) = f_1(\delta)f_2(\delta)\). After that, we introduce the notion of \(S(\Delta)\) actions on \(S(\Gamma)^\Delta : S(\Delta)\times S(\Gamma)^\Delta \rightarrow S(\Gamma)^\Delta\), \((s, f) \mapsto s\cdot f=f^s\), where \(f^s(\delta) = (f\circ s^{-1})(\delta) = (fs^{-1})(\delta)\) for all \(\delta \in \Delta\). Finaly, we give the wreath product \(W\) of \(S(\Gamma)\) by \(S(\Delta)\), and the action of \(W\) on \(\Gamma \times \Delta\).

MSC:

20E22 Extensions, wreath products, and other compositions of groups
20B05 General theory for finite permutation groups
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References:

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