Solvable group

In the history of mathematics, the origins of group theory lie in the search for a proof of the general unsolvability of quintic and higher equations, finally realized by Galois theory. The concept of solvable (or soluble) groups arose to describe a property shared by the automorphism groups of those polynomials whose roots can be expressed using only radicals (square roots, cube roots, etc., and their sums and products).

A group is called solvable if it has a normal series whose factor groups are all abelian.

For finite groups, an equivalent definition is that a solvable group is a group with a composition series whose factors are all cyclic groups of prime order. This is equivalent because every 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.

In keeping with George Polya's dictum that "if there's a problem you can't figure out, there's a simpler problem you can figure out", solvable groups are often useful for reducing a conjecture about a complicated group, into a conjecture about a series of groups with simple structure - cyclic groups of prime order.

All abelian groups are solvable - the quotient A/B will always be abelian if both A and B are abelian. But non-abelian groups may or may not be solvable.

A small example of a solvable, non-abelian group is the symmetric group S₃. In fact, as the smallest simple non-abelian group is A₅, (the alternating group of degree 5) it follows that every group with order less than 60 is solvable.

The group S₅ is not solvable — it has a composition series {E, A₅, S₅}; giving factor groups isomorphic to A₅ and C₂; and A₅ is not abelian. Generalizing this argument, coupled with the fact that A_n is a normal, maximal, non-abelian simple subgroup of S_n for n > 4, we see that S_n is not solvable for n > 4, a key step in the proof that for every n > 4 there are polynomials of degree n which are not solvable by radicals.

The property of solvability is in some senses inheritable, since:

• If G is solvable, and H is a subgroup of G, then H is solvable.
• If G is solvable, and H is a normal subgroup of G, then G/H is solvable.
• If G is solvable, and there is a homomorphism from G onto H, then H is solvable.
• If H and G/H are solvable, then so is G.
• If G and H are solvable, the direct product G × H is solvable.

Supersolvable group

As a strengthening of solvability, a group G is called supersolvable if it has an invariant normal series whose factors are all cyclic; in other words, if it is solvable with each A_i also being a normal subgroup of G, and each A_i+1/A_i is not just abelian, but also cyclic (possibly of infinite order). Since a normal series has finite length by definition, there are uncountable abelian groups which are not supersolvable; but if we restrict ourselves to finite groups, we can consider the following arrangement of classes of groups:

cyclic < abelian < nilpotent < supersolvable < solvable < finite group