Solvable group Totally Explained

Everything about Solvable Group totally explained

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).

Definition

A group is called solvable if it has a normal series whose factor groups are all abelian. Or equivalently, if its derived series, the descending normal series ť

G	riangleright G^ (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 isn't 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 isn't 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's either a prime cyclic or of even order.
Every finite group all whose p-Sylow subgroups are cyclic is a semidirect product of two cyclic groups, in particular solvable. Such groups are called Z-groups.

Properties

Solvability is closed under a number of operations.
Solvable groups form a subvariety of the variety of groups, as they're closed under the taking of homomorphic images, subalgebras and (direct) products:

If G is solvable, and there's a homomorphism from G onto H, then H is solvable; equivalently (by the first isomorphism theorem), if G is solvable, and H is a normal subgroup of G, then G/H is solvable.
If G is solvable, and H is a subgroup of G, then H is solvable.
If G and H are solvable, the direct product G × H is solvable.

Solvability is closed under group extension:

If H and G/H are solvable, then so is G; in particular, if N and H are solvable, their semidirect product is also solvable. It is also closed under wreath product:

If G and H are solvable, and X is a G-set, then the wreath product of G and H with respect to X is also solvable.

Related concepts

Supersolvable groups

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. Since a normal series has finite length by definition, uncountable groups are not supersolvable. In fact, all supersolvable groups are finitely generated, and an abelian group is supersolvable if and only if it's finitely generated. The alternating group A₄ is an example of a finite solvable group that isn't supersolvable.
If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups:
ť cyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group.

Hypoabelian

A solvable group is one whose derived series reaches the trivial subgroup at a finite stage. For an infinite group, the finite derived series may not stabilize, but the transfinite derived series always stabilizes. A group whose transfinite derived series reaches the trivial group is called a hypoabelian group, and every solvable group is a hypoabelian group. The first ordinal α such that G^(α) = G^(α+1) is called the (transfinite) derived length of the group G, and it has been shown that every ordinal is the derived length of some group .

Further Information

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