# 群与图讨论班 Seminars on Groups and Graphs

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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.

Host: 黄弘毅 Hong Yi Huang

Slides