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.
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 An is a normal, maximal, non-abelian simple subgroup of Sn for n > 4, we see that Sn 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 Ai also being a normal subgroup of G, and each Ai+1/Ai 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 grouppl:Grupa rozwiązalna