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2023•Communications in AlgebraRequires access

On the normalizer of an iterated wreath product

Fernando Szechtman

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Abstract

Given a group G and n≥0, let W(G, n) be the associated iterated wreath product—unrestricted when G is infinite—viewed as a permutation group on Gn. We prove that the normalizer of W(G, n) in the symmetric group S(Gn) is equal to Mn⋉W(G,n), where Mn is isomorphic to Aut(G)n. The action of Aut(G)n on W(G, n) is recursively described.Communicated by Mark Lewis

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Given a group G and n≥0, let W(G, n) be the associated iterated wreath product—unrestricted when G is infinite—viewed as a permutation group on Gn. We prove that the normalizer of W(G, n) in the symmetric group S(Gn) is equal to Mn⋉W(G,n), where Mn is isomorphic to Aut(G)n. The action of Aut(G)n on W(G, n) is recursively described.Communicated by Mark Lewis

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Available abstract

Given a group G and n≥0, let W(G, n) be the associated iterated wreath product—unrestricted when G is infinite—viewed as a permutation group on Gn. We prove that the normalizer of W(G, n) in the symmetric group S(Gn) is equal to Mn⋉W(G,n), where Mn is isomorphic to Aut(G)n. The action of Aut(G)n on W(G, n) is recursively described.Communicated by Mark Lewis

Key concepts: Wreath product, Centralizer and normalizer, Mathematics, Combinatorics, Permutation group, Iterated function, Symmetric group, Mathematics Subject Classification

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