On Functional Equations in Linear Topological Spaces

HYERS +-subgroup cannot be commutators arising from two operators whose orders are less than pk+l in view of the fact that the pth power of every operator of G appears in its +-subgroup.The operators of order pk+1 contained in G generate G and hence the operators of highest order in the +)-subgroup of G are commutative with every operator of a set of generators of G. Therefore it results that if a group of order pm has m -k independent generators, p being an odd prime and the 4-subgroup of G being cyclic, then the commutator subgroup of G is of order p whenever G is nonabelian.In particular, there is one and only one non-abelian group of order pm which is generated by two of its operators and has a cyclic 4- subgroup.