Overgroups of a linear Singer cycle in classical groups

Brookfield, Tom (2015). Overgroups of a linear Singer cycle in classical groups. University of Birmingham. Ph.D.

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Abstract

There are well known embeddings of GL $_n$ ( $q$ ) in Sp $_2$ $_n$ ( $q$ ) and GO+ $_2$ $_n$ ( $q$ ), as well as an embedding of GL $_n$ ( $q$ $^2$ ) in GU $_2$ $_n$ ( $q$ ). These give embeddings of Singer cycles of GL $_n$ ( $q$ ) and GL $_n$ ( $q$ $^2$ ) in the corresponding classical group, which we call $linear$ $Singer$ $cycles$ . In this thesis, we classify the overgroups of such a Singer cycle in these groups.