Solvable Group - Definition

Definition

A group is called solvable if it has a subnormal series whose factor groups are all abelian, that is, if there are subgroups such that is normal in, and is an abelian group, for .

Or equivalently, if its derived series, the descending normal series

where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup {1} of G. These two definitions are equivalent, since for every group H and every normal subgroup N of H, the quotient H/N is abelian if and only if N includes H(1). The least n such that is called the derived length of the solvable group G.

For finite groups, an equivalent definition is that a solvable group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a finite group has finite composition length, and every finite simple abelian group is cyclic of prime order. The Jordan–Hölder theorem guarantees that if one composition series has this property, then all composition series will have this property as well. For the Galois group of a polynomial, these cyclic groups correspond to nth roots (radicals) over some field. The equivalence does not necessarily hold for infinite groups: for example, since every nontrivial subgroup of the group Z of integers under addition is isomorphic to Z itself, it has no composition series, but the normal series {0,Z}, with its only factor group isomorphic to Z, proves that it is in fact solvable.