Ghadbane, Nacer Wreath product of permutation groups and their actions on a sets. (English) Zbl 1499.20074 Casp. J. Math. Sci. 10, No. 2, 142-155 (2021). 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 Keywords:group; acts of group in a set; morphism of groups; semi-direct product of groups; wreath product of groups PDFBibTeX XMLCite \textit{N. Ghadbane}, Casp. J. Math. Sci. 10, No. 2, 142--155 (2021; Zbl 1499.20074) Full Text: DOI References: [1] M. R. Adhikari, A. Adhikari, ”Basic Modern Algebra With Applications, Springer (2014). · Zbl 1284.00001 [2] Ibrahim A. A and Audu M. S, “On Wreath Product of Permutation Groups”, Proyecciones, Vol. 26, No1, (2007), pp.73-90. · Zbl 1169.20300 [3] B. Baumslag and B. Chandler. “Theory and Problems of Group Theory”, New York University, (1968). [4] O. Bogopolski, “Introduction to Group Theory”, European Mathematical Society, (2008). · Zbl 1215.20001 [5] W. Chang, ”Image processing with wreath product groups, HARVEY MUDD, (2004). [6] L. Daniels, ”Group Theory and the Rubik’s Cube, Lakehead Univrsity, Canada, (2014). [7] D. Guin et T. Hausberger, “Alg‘ebre Tome 1 Groupes, Corps et Th´eorie de Galois”, EDP Sciences, (2008). · Zbl 1153.12001 [8] J. M. Howie, “Fundamentals of Semigroups Theory”, Oxford Science Publications, (1995). · Zbl 0835.20077 [9] W. Ledermann, “Introduction to Group Theory”, Longman Group Limited, London, (1973). · Zbl 0284.20001 [10] D. L. Kreher. Group Theory Notes, (2012). [11] J. M. Howie, “Fundamentals of Semigroup Theory”, Oxford Science Publications, (1995). · Zbl 0835.20077 [12] A. E. Nagy and C. L. Nehaniv, “Cascade Product of Permutation Groups”, Centre for computer science and informatics, U. K and Centre for research in mathematics, Australia, (2013). [13] Audu M. S, “Wreath Product of Permutation Group”, A Research Oriented Course In Arithmetics of Elliptic Curves, Groups and Loops, Lecture Notes Series, National Mathematical Centre, Abuja, (2001). [14] J. D. P. Meldrum, ”Wreath products of groups and semigroups, Longman, (1995). · Zbl 0833.20001 [15] J. P. Serre, “Trees”, Springer-Velag Berlin Heidelberg New York, (1980). · Zbl 0548.20018 [16] H. Straubing, “Finite Automata, Formal Logic, and Circuit Complexity”, Springer Science+Business Media, LLC, (1994) · Zbl 0816.68086 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.