Solvable group

In the history of mathematics, the origins of group theory lie in the search for a proof of the general insolvability 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).

More generally, in keeping with Polya's dictum that "if there's a problem you can't figure out, there's a simpler problem you can't figure out", solvable groups can often be used to reduce a conjecture about a complicated group, into a conjecture about a series of groups with simple structure - cyclic groups of prime order.

Let E be the trivial subgroup; then a normal series of a group G is a finite sequence of subgroups, E = A₁, A₂, ..., A_i, ..., A_n-1, A_n = G, where each A_i is a normal subgroup of A_i+1. There is no requirement that A_i be a normal subgroup of G (a series with this additional property is called an invariant series); nor is there any requirement that A_i be maximal in A_i+1.

A series with the additional property that A_i ≠ A_i+1 for all i is called a normal series without repetition; equivalently, each A_i is a proper normal subgroup of A_i+1.

If we require that each A_i be a maximal, proper, normal subgroup of A_i+1, it then follows that the factor group A_i+1 / A_i will be simple in each case. This gives the following definition: a composition series of a group is a normal series, without repetition, where the factors A_i+1 / A_i are all simple.

There are no additional subgroups which can be "inserted" into a composition series; and it can be seen that, if a composition series exists for a group G, then any normal series of G can be refined to a composition series, informally, by inserting subgroups into the series up to maximality. Every finite group has a composition series; but not every infinite group has one (for example, the additive group of integers (Z, +) has no composition series).

In general, a group will have multiple, different composition series. For example, the cyclic group C₁₂ has {E, C₂, C₆, C₁₂}, {E, C₂, C₄, C₁₂}, and {E, C₃, C₆, C₁₂} as different composition series. However, the result of the Jordan-H�lder Theorem is that any two composition series of a group are equivalent, in the sense that the sequence of factor groups in each series are the same, up to rearrangement of their order in the sequence A_i+1 / A_i. In the above example, the factor groups are isomorphic to {C₂, C₃, C₂}, {C₂, C₂, C₃}, and {C₃, C₂, C₂}, respectively.

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

For finite groups, it is equivalent (and useful) to require that a solvable group have a composition series whose factors are all cyclic of prime order (as every simple, abelian group must be cyclic of prime order). The Jordan-H�lder 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.

Certainly, any abelian group will be solvable - the quotient A/B will always be abelian if both A and B are abelian. The situation is not always so clear in the case of non-abelian groups.

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 however 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 rather '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.

As a strengthening of solvability, a group G is called supersolvable if it 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