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Basic notions in group theory
category of groups
subgroups, normal subgroups
quotient groups
group homomorphisms, kernel, image
(semi-)direct product, direct sum
types of groups
finite, infinite
discrete, continuous
multiplicative, additive
abelian, cyclic, simple, solvable

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. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, one can often define a direct product of objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true In Mathematics, a quintic equation is a Polynomial Equation of degree five In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field 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. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, an n th root of a Number a is a number b such that bn = a. , and their sums and products).

Contents

Definition

A group is called solvable if it has a normal series whose factor groups are all abelian. In Mathematics, a subgroup series is a chain of Subgroups 1 = A_0 \leq A_1 \leq \cdots \leq A_n = G In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Or equivalently, if its derived series, the descending normal series

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

where every subgroup is the commutator subgroup of the previous one, ever reaches the trivial subgroup {1} of G. In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup 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, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. ↔ The least n such that G(n) = {1} 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 series whose factors are all cyclic groups of prime order. In Abstract algebra, a composition series provides a way to break up an algebraic structure such as a group or a module, into simple pieces In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Group theory, a branch of Mathematics, the term order is used in two closely related senses the order of a group is This is equivalent because a finite abelian group has finite composition length, and every finite simple abelian group is cyclic of prime order. SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning The Jordan-Hölder theorem guarantees that if one composition series has this property, then all composition series will have this property as well. In Abstract algebra, a composition series provides a way to break up an algebraic structure such as a group or a module, into simple pieces 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 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

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 (b December 13, 1887 &ndash d September 7, 1985, in Hungarian Pólya György) was a Hungarian

Examples

All abelian groups are solvable - the quotient A/B will always be abelian if A is abelian. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G But non-abelian groups may or may not be solvable.

More generally, all nilpotent groups are solvable. In Group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the In particular, finite p-groups are solvable, as all finite p-groups are nilpotent. In Mathematics, given a Prime number p, a p -group is a Periodic group in which each element has a power of p In Mathematics, given a Prime number p, a p -group is a Periodic group in which each element has a power of p

A small example of a solvable, non-nilpotent group is the symmetric group S3. In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying 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, 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 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. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

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

Every finite group all whose p-Sylow subgroups are cyclic is a semidirect product of two cyclic groups, in particular solvable. In Mathematics, specifically Group theory, the Sylow theorems, named after Ludwig Sylow, form a partial converse to Lagrange's theorem, which Such groups are called Z-groups. In Mathematics, especially in the area of Algebra known as Group theory, the term Z-group refers to a number of distinct types of groups:

Properties

Solvability is closed under a number of operations.

Solvable groups form a subvariety of the variety of groups, as they are closed under the taking of homomorphic images, subalgebras and (direct) products:

Solvability is closed under group extension:

It is also closed under wreath product:

Related concepts

Supersolvable groups

Main article: 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 Mathematics, a group is supersolvable (or supersoluble) if it has an invariant Normal series where all the factors are cyclic groups 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. In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the 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:

cyclic < abelian < nilpotent < supersolvable < polycyclic < solvable < finitely generated group. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a nilpotent group is a group having a special property that makes it "almost" abelian, through repeated application of the In Mathematics, a group is supersolvable (or supersoluble) if it has an invariant Normal series where all the factors are cyclic groups In Mathematics, especially in the area of Abstract algebra known as Group theory, a polycyclic group is a Solvable group that satisfies the maximal In Abstract algebra, a generating set of a group G is a Subset S such that every element of G can be expressed as the

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. In Mathematics, in the field of Group theory, the perfect core (or perfect radical) of a group is its largest perfect Subgroup 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 (Malcev 1949).

References

External links

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