9 research outputs found
The -closure of a solvable permutation group is solvable
Let be a positive integer and let be a finite set. The
-closure of is the largest permutation
group on having the same orbits as in its induced action on the
Cartesian product . The -closure and -closure of a solvable
permutation group need not be solvable. We prove that the -closure of a
solvable permutation group is always solvable for
On finite totally -closed groups
An abstract group is called totally -closed if for any set with , where is the largest subgroup of whose orbits on are the same orbits of . In this paper, we classify the finite soluble totally -closed groups. We also prove that the Fitting subgroup of a totally -closed group is a totally -closed group. Finally, we prove that a finite insoluble totally -closed group of minimal order with non-trivial Fitting subgroup has shape , with cyclic, and is a finite group with a unique minimal normal subgroup, which is nonabelian