Complete finite Frobenius groups and wreath products

A group G is complete if it has a trivial center and for every automorphism ϕ of G there is an element x ∈ G such that ϕ(g) = x -1gx for all g ∈ G. H. Wielandt and J. S. Rose proved that every finite group G can be embedded as a subnormal subgroup in a finite complete group K, meaning that there exists a sequence of subgroups {G i} of K such that G=G0⊲ G1⊲ &cdots;⊲G i⊲&cdots; ⊲Gn =K. Moreover if G is solvable then K can be solvable too. This dissertation classifies complete Frobenius groups and complete finite permutational wreath products, in addition to investigating the structure of an odd-order complete group.;We also show that in a finite permutational wreath product G &m22; H, if the base group is not characteristic then G is the semidirect product of an odd-order abelian group of index 2 with a cyclic group of order 2 acting by inversion. In the case where H acts transitively we provide a biconditional statement and determine that H is also a wreath product, which confirms earlier results by P. Neumann and Y. V. Bodnarchuk. Lastly we investigate the structure of the automorphism group of G &m22; H when the base is not characteristic.