Minimal degrees of quotient groups

Abstract

For a finite group G, the minimal faithful permutation representation degree, denoted by m(G), is defined as the smallest n ∈ {0, 1, 2, . . .} such that G embeds in Sym(n). The task of determining m(G) for an arbitrary G is a complex undertaking, and can be linked to addressing a difficult minimisation problem concerning the lattice of subgroups of G.

It is interesting to note that the relationship between the minimal degrees of quotient groups and their parent groups is quite uncertain. Despite the fact that the quotient group may be simpler than the parent group, its lattice of subgroups may be more restrictive, so that, when solving the minimisation problem, the minimal degree of the quotient group can actually be greater than the minimal degree of the parent group. In such cases, the parent group is called exceptional. Though exceptional groups are not particularly rare, this terminology, introduced in the 1980s, has persisted.

In this dissertation, we study the delicate relationship between the minimal degrees of finite groups and their respective quotient groups. We address some gaps in the current literature, rectify some existing flaws, and introduce new terminologies and directions for future research. The thesis is a blend of mathematical argument and concrete examples, supported by the use of computer algebra software.