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Orientably-Regular \(\pi\)-Maps and Regular \(\pi\)-Maps

田瑶 Yao Tian

Captial Normal University (首都师范大学)

Time: 16:00-17:00 (GMT+8), Tuesday October 17th, 2023
Location: Zoom

Abstract: Let \(\mathcal{M}\) be an orientably-regular (resp. regular) map with the number \(n\) vertices. By \(G^+\) (resp. \(G\)) we denote the group of all orientation-preserving automorphisms (resp. all automorphisms) of \(\mathcal{M}\). Let \(\pi\) be the set of prime divisors of \(n\). A Hall \(\pi\)-subgroup of \(G^+\) (resp. \(G\)) is meant a subgroup such that the prime divisors of its order all lie in \(\pi\) and the primes of its index all lie outside \(\pi\). If the number \(n\) is a prime \(p\)-power, then the map is called a \(p\)-map. An orientably-regular (resp. A regular ) \(p\)-map is called \(\it{solvable}\) if the group \(G^+\) of all orientation-preserving automorphisms (resp. the group \(G\) of automorphisms) is solvable; and called \(\it {normal}\) if \(G^+\) (resp. \(G\)) contains the normal Sylow \(p\)-subgroup. In this talk, I will outline some important background information and progress towards \(\pi\)-maps, especially, \(p\)-maps. For \(p\)-maps, we have proved that they are solvable, moreover, they are normal for very few exceptions. Along this way, the solvability and normality of \(\pi\)-maps are given under certain conditions.

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