## 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

### Citations:

Zbl 1371.16037; Zbl 1441.12001; Zbl 1498.16043; Zbl 1418.20008

GAP
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### References:

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