Regular orbits of cyclic subgroups of the simple classical groups

Emmett, Lynn (2005) Regular orbits of cyclic subgroups of the simple classical groups. Doctoral thesis, University of East Anglia.

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Abstract

Let C be a group contained in Sym(O.) , and let C be a cyclic subgroup of G. Then the question we ask is: for which groups G C Sym(O.) does every cyclic subgroup C have a regular orbit on O.? In this thesis we look at the simple symplectic, orthogonal and unitary groups of rank three. Let V be a vector space such that V = F~1 and let I (V) be a symplectic, orthogonal or unitary group. We denote by I (V)' the commutator subgroup of I (V). Then the main result contained here is the following:
Let G be a group such that I(V )' ~ G ~ J(V) with the conditions that J(V) =I 0 ±(2, q) , o+(4, q) and q > 5. Let n be a non-trivial, transitive C-set with kernel contained in Z(G). Then C has an orbit of size IC/(C n Z(C))I on n.

If C has an orbit of size k = IC/ (Cn Z(C))I on n then there exists an w En such that ICwl = k. Thus if cw= w for some c E C then c E Z(G). Hence an easy corollary to the main result is:
Let C be a simple symplectic, orthogonal or unitary group over a field of more than 5 elements. Then every cyclic subgroup of C has a regular orbit in every non-trivial C-set n.

Item Type:	Thesis (Doctoral)
Faculty \ School:	Faculty of Science
Depositing User:	Chris White
Date Deposited:	24 Aug 2026 13:13
Last Modified:	24 Aug 2026 13:13
URI:	https://ueaeprints.uea.ac.uk/id/eprint/104305
DOI:

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