| This thesis addresses the question of which subgroups G of the finite symmetric group Sn contain an n-cycle.; The solution to the case n = p a prime is based on a theorem of Burnside. When n is not prime, we apply a theorem based on results of Burnside and Schur to see that the groups G we are looking for are either imprimitive or 2-transitive. Using the Classification of Finite Simple Groups, we look at each possible group G and determine which do indeed contain an n-cycle.; We give a list of subgroups, maximal in Sn or in An, containing an n-cycle: (1) An, n odd; (2) Sn/mwrSm (with the standard action), for all m dividing n with 1 < m < n; (3) PGammaL( d, q), for all d, q with n = (qd - 1)/( q - 1); (4) AGL(1, p), for n = p a prime; (5) PSL(2, 11) (in an exceptional action), for n = 11; (6) M11, for n = 11; (7) M23, for n = 23.; We also discuss some of this problem's applications. |
