Remarks on Frobenius Groups

Abstract

Let the finite group G act transitively and non-regularly on a finite set whose cardinality |Ω| is greater than one. Use N to denote the full set of fixed-point-free elements of G acting on along with the identity element. Write H to denote the stabilizer of some α ∈ Ω in G. In the note, it is proved that the subset N is a subgroup of G if and only if G is a Frobenius group. It is also proved G = {N}H, where {N} is the subgroup of G generated by N.