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The mutually normalizing regular subgroups of the holomorph of a cyclic group of prime power order. (English) Zbl 1510.20004

Let \(G\) be a group and let \(\mathrm{Sym}(G)\) denote its symmetric group. The normalizing graph of \(G\) is the undirected graph whose vertices are the regular subgroups of \(\mathrm{Sym}(G)\), where two of them are joined by an edge whenever they mutually normalize each other. The local normalizing graph of \(G\) is the subgraph obtained by restricting to the regular subgroups which lie inside the holomorph of \(G\).
The normalizing graph is of interest because of its connections with skew brace, an algebraic object defined in [L. Guarnieri and L. Vendramin, Math. Comput. 86, 2519–2534 (2017; Zbl 1371.16037)] as a tool to study solutions to the Yang-Baxter equation. Briefly speaking, any regular subgroup \(N\) of \(\mathrm{Sym}(G)\) induces a group operation \(\circ_N\) on \(G\) via transport, and an edge with endpoints \(N_1,N_2\) in the normalizing graph corresponds to the bi-skew brace \((G,\circ_{N_1},\circ_{N_2})\) in the sense of [L. N. Childs, New York J. Math. 25, 574–588 (2019; Zbl 1441.12001)]. In particular, a complete subgraph of the normalizing graph is equivalent to a brace block in the sense of [A. Koch, J. Pure Appl. Algebra 226, Article ID 107047, 15 p. (2022; Zbl 1498.16043)].
The paper under review determines the local normalizing graph of any cyclic group of prime power order \(p^n\). It employs the method of gamma functions as introduced by [A. Caranti and F. Dalla Volta, J. Algebra 507, 81–102 (2018; Zbl 1418.20008)]. As usual, the cases when \(p\geq 3\) and \(p=2\) require slightly different treatments.

MSC:

20B35 Subgroups of symmetric groups

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References:

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