Solvable group

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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. Or equivalently, if the descending normal series

$G\triangleright G^{(1)}\triangleright G^{(2)} \triangleright \cdots,$

where every subgroup is the derived subgroup of the previous one, ever 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⁾.

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. 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.

In keeping with George Pólya'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: abelian groups (and in the finite case, cyclic groups of prime order).

1 Examples
2 Properties
3 Supersolvable group
4 External links

[edit] Examples

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

More generally, all nilpotent groups are solvable. In particular, finite p-groups are solvable, as all finite p-groups are nilpotent.

A small example of a solvable, non-nilpotent 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} (and the Jordan-Hölder theorem states that every other composition series is equivalent to that one), 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 celebrated Feit–Thompson theorem states that every finite group of odd order is solvable. In particular this implies that if a finite group is simple, it is either a prime cyclic or of even order.

[edit] Properties

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.

[edit] Supersolvable group

As a strengthening of solvability, a group G is called supersolvable (or supersoluble) 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, uncountable abelian groups are not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it is finitely generated.