On one-sided stabilizers of subsets of finite groups

It was initiated by the second author to investigate in which groups the left and right stabilizers of subsets have equal order. First we prove that if the left and right stabilizers of sets of prime power size are equal order then the group is supersolvable. We also characterize those 2-groups which satisfy this property for p = 2. We show that if in a finite group, the left and right stabilizers of sets of prime power size have equal order, then the commutator subgroup is abelian. Finally we characterize hamiltonian groups with the help of one-sided stabilizers. Received: 18 April 2005; revised 11 May 2005