On the automorphism groups of regular maps. (English) Zbl 1535.20106
Let \(\mathcal{M}\) be an orientably regular (regular) map with \(n\) vertices and let \(G^{+}=\operatorname{Aut}^{+}(\mathcal{M})\) (\(G=\operatorname{Aut}(\mathcal{M})\)) the group of all orientation-preserving automorphisms (resp. all automorphisms) of \(\mathcal{M}\). Let \(\pi(n)\) be the set of prime divisors of \(n\).
In the paper under review, the authors prove the following results: (1) let \(\mathcal{M}\) be an orientably regular map where \(n\) is odd, then \(G^{+}\) is solvable and contains a normal Hall \(\pi(n)\)-subgroup; (2) let \(\mathcal{M}\) be a regular map where \(n\) is odd, then \(G\) is solvable if it has no composition factors isomorphic to \(\mathrm{PSL}(2,q)\) for any odd prime power \(q\not = 3\) and \(G\) contains a normal Hall \(\pi\)-subgroup if and only if it has a normal Hall \(2'\)-subgroup.
In the paper under review, the authors prove the following results: (1) let \(\mathcal{M}\) be an orientably regular map where \(n\) is odd, then \(G^{+}\) is solvable and contains a normal Hall \(\pi(n)\)-subgroup; (2) let \(\mathcal{M}\) be a regular map where \(n\) is odd, then \(G\) is solvable if it has no composition factors isomorphic to \(\mathrm{PSL}(2,q)\) for any odd prime power \(q\not = 3\) and \(G\) contains a normal Hall \(\pi\)-subgroup if and only if it has a normal Hall \(2'\)-subgroup.
Reviewer: Egle Bettio (Venezia)
MSC:
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 05E18 | Group actions on combinatorial structures |
Software:
MagmaReferences:
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