On the solvable, nilpotent, and supersolvable groups of order at most two hundred

The emphasis of this paper is to determine whether a group is solvable (resp., nilpotent, supersolvable) based on its order. Throughout the thesis, a number is considered solvable (resp., nilpotent, supersolvable) if every group of that particular order is solvable (resp., nilpotent, supersolvable). Using many of the standard results for solvable groups as a blueprint, we focus primarily on finite nilpotent groups. There are three main results established in the paper. First, a finite group is nilpotent if and only if every Sylow subgroup is normal (unique). Next, a group of order p²q, for primes p < q, is nilpotent whenever p q - 1. The same principle does not hold true if p > q. And finally, the dihedral group, D_n, is nilpotent if and only if n is a power of two.