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# Rank 3 permuation characters and maximal subgroups

Tong Viet, Hung Phi (2009)
Ph.D. thesis, University of Birmingham.

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## Abstract

Let G be a transitive permutation group acting on a finite set E. Let P be a stabilizer in G of a point in E. We say G is primitive rank 3 on E if P is maximal in G and P has exactly three orbits on E. For any subgroup H of G, we denote by 1 $$\frac{G}{H}$$ the permutation character or permutation module over the complex number field of G on the cosets G/H. Let H and K be subgroups of G. We say 1 $$\frac{G}{H}$$ $$\leq$$ 1$$\frac{G}{K}$$if 1 $$\frac{G}{K}$$ $$\leq$$ -1$$\frac{G}{H}$$is either 0 or a character of G. Also a finite group G is called nearly simple primitive rank 3 on E if there exists a quasi-simple group L such that L/Z(L) $$\triangleleft$$ G/Z(L) $$\leq$$ Aut(L/Z(L)) and G acts as a primitive rank 3 permutation group on some cosets of a subgroup of L. In this thesis we classify all maximal subgroups M of a class of nearly simple primitive rank 3 groups G acting on E such that 1 $$\frac{G}{H}$$ $$\leq$$ 1 $$\frac{G}{H}$$ where P is a stabilizer of a point in E. This result has an application to the study of minimal genus of algebraic curves which admit group actions.

Type of Work: Ph.D. thesis. Magaard, Kay (Dr) Schools (1998 to 2008) > School of Mathematics & Statistics Pure Mathematics QA Mathematics University of Birmingham 290
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