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	Topics Solvable group Topic Home In the history of mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , the origins of group theoryGroup theory Group theory is that branch of mathematics concerned with the study of groups.... lie in the search for a proofMathematical proof In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true.... of the general unsolvability of quintic and higher equations, finally realized by Galois theoryGalois theory In mathematics, more specifically in abstract algebra, Galois theory, named after variste Galois, provides a connection betw... . The concept of solvable (or soluble) groups arose to describe a property shared by the automorphism groups of those polynomialsPolynomial In mathematics, a polynomial is an expression in which a finite number of constants and variables are combined using only ad... 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 seriesNormal series In mathematics, a normal series of a group G is a sequence of subgroups, each a normal subgroup of the next one.... whose factor groups are all abelianAbelian group In mathematics, an abelian group, also called a commutative group, is a group such that a * b = b * a... . Or equivalently, if its derived series, the descending normal series where every subgroup is the commutator subgroupCommutator subgroup In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the s... of the previous one, ever reaches the trivial subgroup of G. These two definitions are equivalent, since for every group H and every normal subgroupNormal subgroup In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup.... N of H, the quotient H/N is abelian if and only ifIf and only if In logic and fields that rely on it, such as mathematics and philosophy, "if and only if" is a logical connective between s... N includes H⁽¹⁾. The least n such that is called the derived length of the solvable group G. For finite groups, an equivalent definition is that a solvable group is a group with a composition seriesComposition series In mathematics, a composition series of a group G is a normal series... whose factors are all cyclic groups of primePrime number In mathematics, a prime number is a natural number that has exactly two natural number divisors, which are 1 and the prime... orderOrder (group theory) In group theory, a branch of mathematics, the term *order* is used in two closely related senses:... . This is equivalent because a finite abelian group has finite composition length, and every finite simpleSimple group In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial gro... 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 isomorphicGroup isomorphism In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between t... to Z itself, it has no composition series, but the normal series , with its only factor group isomorphic to Z, proves that it is in fact solvable. In keeping with George PólyaGeorge Pólya George Plya was a mathematician, who was born in Budapest, Hungary and died in Palo Alto, USA.... '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). Examples All abelian groups are solvable - the quotientQuotient group In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G'... A/B will always be abelian if A is abelian. But non-abelian groups may or may not be solvable. More generally, all nilpotent groupNilpotent group In group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated ap... s are solvable. In particular, finite p-groupsFacts About P-group In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as ... are solvable, as all finite p-groupsP-group In mathematics, given a prime number p, a p-group is a periodic group in which each element has a power of p as ... are nilpotent. A small example of a solvable, non-nilpotent group is the symmetric groupSymmetric group In mathematics, the symmetric group on a set X, denoted by SX or Sym, is the group whose underlying set is the set o... S₃. In fact, as the smallest simple non-abelian group is A₅, (the alternating groupAlternating group In mathematics, an alternating group is the group of even permutations of a finite set.... 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 (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 polynomialPolynomial In mathematics, a polynomial is an expression in which a finite number of constants and variables are combined using only ad... s of degree n which are not solvable by radicals. The celebrated Feit–Thompson theoremFeit–Thompson theorem In mathematics, the Feit?Thompson theorem, or odd order theorem, states that every finite group of odd order is solvab... 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. 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-groupZ-group In mathematics, especially in the area of algebra known as group theory, the term Z-group refers to a number of distinct typ... s. Properties Solvability is closed under a number of operations. Solvable groups form a subvarietyVariety (universal algebra) In universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given... of the variety of groups, as they are closed under the taking of homomorphicHomomorphism Overview In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures .... images, subalgebraSubalgebra In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra ... s and (direct) productsDirect product Overview In mathematics, one can often define a direct product of objects... : If G is solvable, and there is a homomorphismGroup homomorphism In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that fo... from G onto H, then H is solvable; equivalently (by the first isomorphism theoremIsomorphism theorem In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the e... ), if G is solvable, and N is a normal subgroup of G, then G/N is solvable. The previous property can be expanded into the following property: G is solvable if and only if both N and G/N are solvable. If G is solvable, and H is a subgroup of G, then H is solvable. If G and H are solvable, the direct productDirect product In mathematics, one can often define a direct product of objects... G × H is solvable. Solvability is closed under group extensionGroup extension Overview In mathematics, for G a group, G′ is an extension of G if there is an exact sequence... : If H and G/H are solvable, then so is G; in particular, if N and H are solvable, their semidirect productSemidirect product Summary In group theory, a semidirect product describes a particular way in which a group can be put together from two subgroups, on... 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 productWreath product In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect 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 is finitely generated. The alternating group A₄ is an example of a finite solvable group that is not supersolvable. If we restrict ourselves to finitely generated groups, we can consider the following arrangement of classes of groups: cyclicCyclic group In group theory, a cyclic group or monogenous group is a group that can be generated by a single element, in the sense... < abelianAbelian group In mathematics, an abelian group, also called a commutative group, is a group such that a * b = b * a... < nilpotentNilpotent group In group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated ap... < supersolvableSupersolvable group In mathematics, in the field of group theory, a group is said to be supersolvable if it has a chief series where all the chief fac... < polycyclicFacts About Polycyclic group In mathematics and group theory, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups.... < 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 . External links - orders of non-solvable finite groups.