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). Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Group theory is that branch of mathematics concerned with the study of groups. ... Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 Quintic functions are polynomial functions in which the highest degree is five. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, an nth root of a number a is a number b, such that bn=a. ...

A group is called solvable if it has a normal series whose factor groups are all abelian. Or equivalently, if the descending normal series In mathematics, a normal series of a group G is a sequence of subgroups, each a normal subgroup of the next one. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

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⁾. In mathematics, the derived group (or commutator subgroup) of a group G is the subgroup G1 generated by all the commutators of elements of G; that is, G1 = <[g,h] : g,h in G>. Note that the set of all commutators of the group is, generally, not a group (in... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... This article does not cite any references or sources. ...

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 mathematics, a composition series of a group G is a chain of subgroups of G satisfying where stands for normal subgroup, such that each quotient group Hi+1/Hi is a simple group. ... In mathematics, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that every element of the group is a power of a. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In group theory, the term order is used in two closely related senses: the order of a group is its cardinality, i. ... In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. ... In mathematics, a composition series of a group G is a normal series such that each Hi is a maximal normal subgroup of Hi+1. ... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...

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). George Pólya (December 13, 1887 – September 7, 1985, in Hungarian Pólya György) was a Hungarian mathematician. ...

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. In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ...

More generally, all nilpotent groups are solvable. In particular, finite p-groups are solvable, as all finite p-groups are nilpotent. In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ... In mathematics, given a prime number p, a p-group is a group in which each element has a power of p as its order. ... In mathematics, given a prime number p, a p-group is a group in which each element has a power of p as its order. ...

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. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics an alternating group is the group of even permutations of a finite set. ...

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. In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

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. In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. ...

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.

Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ...

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.

If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups:

cyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group

In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In group theory, a nilpotent group is a group having a special property that makes it almost abelian, through repeated application of the commutator operation, [x,y] = x-1y-1xy. ... In mathematics and group theory, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). ... In abstract algebra, a generating set of a group is a subset S such that every element of G can be expressed as the product of finitely many elements of S and their inverses. ...

External links

• A056866 - orders of non-solvable finite groups.