A group having a finite subnormal series with Abelian quotient groups (see Subgroup series). It also possesses a normal series with Abelian quotient groups (such series are called solvable). The length of the shortest solvable series of the group is called its derived length or degree of solvability. The most important of these series is the commutator series or the derived series (see Commutator subgroup of a group). The term "solvable group" arose in Galois theory in connection with the solvability of algebraic equations by radicals.
Finite solvable groups have subnormal series with quotient groups of prime order. These groups are characterized by the following converse to Lagrange's theorem: For any factorization $n=n_1n_2$ of the order $n$ of a group into two relatively prime factors there exists a subgroup of order $n_1$, and any two subgroups of order $n_1$ are conjugate. If the order of a finite group is divisible by two prime numbers only, then the group is solvable. In the class of solvable groups the finite groups are distinguished as the finitely-generated periodic groups.
Particular cases of solvable groups are nilpotent groups, polycyclic groups and meta-Abelian groups (cf. Nilpotent group; Polycyclic group; Meta-Abelian group). The finitely-generated groups which are extensions of an Abelian normal subgroup by a polycyclic quotient group form an important subclass. They satisfy the maximum condition for normal subgroups (see Chain condition) and are residually finite (see Residually-finite group). Every connected solvable Lie group (and also every solvable group of matrices, which is connected in the Zariski topology) has a nilpotent commutator subgroup. Every solvable matrix group over an algebraically closed field has a subgroup of finite index conjugate to a subgroup of the triangular group (see Lie–Kolchin theorem).
The set of all solvable groups of length not exceeding $l$ forms a variety (see Variety of groups). The free groups of such varieties are called free solvable groups.
|||A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian)|
|||M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)|
See also Burnside problem 1).
|[a1]||D.J.S. Robinson, "Finiteness condition and generalized soluble groups" , 1–2 , Springer (1972)|
Solvable group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Solvable_group&oldid=34070