A survey of finite solvable groups

The study of solvable groups was initiated by Evariste Galois (1811-1832) who gave a necessary and sufficient condition for the solvability of an equation of any degree by radicals, i.e., a polynomial is solvable by radicals if and only if its Galois group Gal(R/F) is a solvable group. More recently, Philip Hall proved many new results on finite solvable groups including the famous generalization of Sylow's Theorem. This paper looks at finite solvable groups in terms of the normal structure as well as the arithmetic, or Sylow, structure. Also, some sufficient conditions for solvability are given, as well as some further generalizations of Sylow's Theorem. All groups considered here are finite.