Transversals in Groups with an application to loops

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The aim of the project is to study transversals of subgroups in a group G. We plan to investigate two types of questions: (a) Suppose a certain type of transversal exists - for instance a normal or global transversal - for a subgroup of G, or for each subgroup of G or a special class of subgroups of G. What is the implication of this on the structure of G?(b) Let G be a certain type of group - for instance primitive, supersoluble or soluble - which type of transversal has to exist?In particular, we plan to continue the classification of the simple Bol loops, which is a special case of question (a), as well as the classification of the primitive permutation groups which possess a regular subgroup.