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

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