# 群与图讨论班 Seminars on Groups and Graphs

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# Irredundant bases for the primitive actions of the symmetric and the alternating groups

## 吴培然 Peiran Wu

University of St Andrews

Time: 16:00-18:00 (GMT+8), Monday October 24th, 2022
Location: Zoom

Abstract: A finite transitive permutation group $$G$$ on a set $$\Omega$$ is primitive if and only if any point stabiliser is a maximal subgroup of $$G$$. The maximum irredundant base size of $$G$$, denoted by $$I(G)$$, is equal to the maximum length of a subgroup chain in $$G$$ consisting of pointwise stabilisers of subsets of $$\Omega$$. In this talk, we consider the case where $$G$$ is primitive and isomorphic to the symmetric or the alternating group of some finite degree. We briefly discuss other statistics of $$G$$ such as the base size and the height, but focus on the behaviour of $$I(G)$$. We deploy the O'Nan–Scott theorem, which classifies the maximal subgroups of such groups $$G$$. The exact value of $$I(G)$$ is known in the intransitive case of the theorem, and “good” upper and lower bounds have been found in the affine case and the primitive wreath case, whereas there is work in progress on the remaining cases.

Host: 黄弘毅 Hong Yi Huang

Slides