Solvable group

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup.

Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable by radicals if and only if the corresponding Galois group is solvable.

Definition

A group G is called solvable if it has a subnormal series whose factor groups (quotient groups) are all abelian, that is, if there are subgroups 1 = G0 < G1 < ⋅⋅⋅ < Gk = G such that Gj−1 is normal in Gj, and Gj/Gj−1 is an abelian group, for j = 1, 2, …, k.

Or equivalently, if its derived series, the descending normal series

where every subgroup is the commutator subgroup of the previous one, eventually reaches the trivial subgroup 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). 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 all of whose factors are cyclic groups of prime order. This is equivalent because a finite group has finite composition length, and 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.

Examples

All abelian groups are trivially solvable – a subnormal series being given by just the group itself and the trivial group. But non-abelian groups may or may not be solvable.

More generally, all nilpotent groups are solvable. In particular, finite p-groups are solvable, as all finite p-groups are nilpotent.

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.

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. This is a key step in the proof that for every n > 4 there are polynomials of degree n which are not solvable by radicals (Abel–Ruffini theorem). This property is also used in complexity theory in the proof of Barrington's theorem.

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.

Any finite group whose p-Sylow subgroups are cyclic is a semidirect product of two cyclic groups, in particular solvable. Such groups are called Z-groups.

Numbers of solvable groups with order n are (start with n = 0)

0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 12, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, ... (sequence A201733 in the OEIS)

Orders of non-solvable groups are

60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1020, 1080, 1092, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, ... (sequence A056866 in the OEIS)

Properties

Solvability is closed under a number of operations.

  • If G is solvable, and there is a homomorphism from G onto H, then H is solvable; equivalently (by the first isomorphism theorem), if G is solvable, and N is a normal subgroup of G, then G/N is solvable.[1]
  • 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.[2]
  • 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.

For any positive integer N, the solvable groups of derived length at most N form a subvariety of the variety of groups, as they are closed under the taking of homomorphic images, subalgebras, and (direct) products. The direct product of a sequence of solvable groups with unbounded derived length is not solvable, so the class of all solvable groups is not a variety.

Burnside's theorem

Burnside's theorem states that if G is a finite group of order paqb where p and q are prime numbers, and a and b are non-negative integers, then G is 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:

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

Virtually solvable groups

A group G is called virtually solvable if it has a solvable subgroup of finite index. This is similar to virtually abelian. Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1.

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 (Malcev 1949).

See also

Notes

  1. ^ Rotman (1995), Theorem 5.16, p. 102, at Google Books
  2. ^ Rotman (1995), Theorem 5.15, p. 102, at Google Books

References

  • Malcev, A. I. (1949), "Generalized nilpotent algebras and their associated groups", Mat. Sbornik N.S., 25 (67): 347–366, MR 0032644
  • Rotman, Joseph J. (1995), An Introduction to the Theory of Groups, Graduate Texts in Mathematics, 148 (4 ed.), Springer, ISBN 978-0-387-94285-8

External links

Almost simple group

In mathematics, a group is said to be almost simple if it contains a non-abelian simple group and is contained within the automorphism group of that simple group: if it fits between a (non-abelian) simple group and its automorphism group. In symbols, a group A is almost simple if there is a simple group S such that

Brauer's theorem on induced characters

Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, which is, in turn, part of the representation theory of a finite group. Let G be a finite group and let Char(G) denote the subring of the ring of complex-valued class functions of G consisting of integer combinations of irreducible characters. Char(G) is known as the character ring of G, and its elements are known as virtual characters (alternatively, as generalized characters, or sometimes difference characters). It is a ring by virtue of the fact that the product of characters of G is again a character of G. Its multiplication is given by the elementwise product of class functions.

Brauer's induction theorem shows that the character ring can be generated (as an abelian group) by induced characters of the form , where H ranges over subgroups of G and λ ranges over linear characters (having degree 1) of H.

In fact, Brauer showed that the subgroups H could be chosen from a very restricted collection, now called Brauer elementary subgroups. These are direct products of cyclic groups and groups whose order is a power of a prime.

Using Frobenius reciprocity, Brauer's induction theorem leads easily to his fundamental characterization of characters, which asserts that a complex-valued class function of G is a virtual character if and only if its restriction to each Brauer elementary subgroup of G is a virtual character. This result, together with the fact that a virtual character θ is an irreducible character if and only if θ(1) > 0 and (where is the usual inner product on the ring of complex-valued class functions) gives a means of constructing irreducible characters without explicitly constructing the associated representations.

An initial motivation for Brauer's induction theorem was application to Artin L-functions. It shows that those are built up from Dirichlet L-functions, or more general Hecke L-functions. Highly significant for that application is whether each character of G is a non-negative integer combination of characters induced from linear characters of subgroups. In general, this is not the case. In fact, by a theorem of Taketa, if all characters of G are so expressible, then G must be a solvable group (although solvability alone does not guarantee such expressions- for example, the solvable group SL(2,3) has an irreducible complex character of degree 2 which is not expressible as a non-negative integer combination of characters induced from linear characters of subgroups). An ingredient of the proof of Brauer's induction theorem is that when G is a finite nilpotent group, every complex irreducible character of G is induced from a linear character of some subgroup.

A precursor to Brauer's induction theorem was Artin's induction theorem, which states that |G| times the trivial character of G is an integer combination of characters which are each induced from trivial characters of cyclic subgroups of G. Brauer's theorem removes the factor |G|, but at the expense of expanding the collection of subgroups used. Some years after the proof of Brauer's theorem appeared, J.A. Green showed (in 1955) that no such induction theorem (with integer combinations of characters induced from linear characters) could be proved with a collection of subgroups smaller than the Brauer elementary subgroups.

The proof of Brauer's induction theorem exploits the ring structure of Char(G) (most proofs also make use of a slightly larger ring, Char*(G), which consists of -combinations of irreducible characters, where ω is a primitive complex |G|-th root of unity). The set of integer combinations of characters induced from linear characters of Brauer elementary subgroups is an ideal I(G) of Char(G), so the proof reduces to showing that the trivial character is in I(G). Several proofs of the theorem, beginning with a proof due to Brauer and John Tate, show that the trivial character is in the analogously defined ideal I*(G) of Char*(G) by concentrating attention on one prime p at a time, and constructing integer-valued elements of I*(G) which differ (elementwise) from the trivial character by (integer multiples of) a sufficiently high power of p. Once this is achieved for every prime divisor of |G|, some manipulations with congruences and algebraic integers, again exploiting the fact that I*(G) is an ideal of Ch*(G), place the trivial character in I(G). An auxiliary result here is that a -valued class function lies in the ideal I*(G) if its values are all divisible (in ) by |G|.

Brauer's induction theorem was proved in 1946, and there are now many alternative proofs. In 1986, Victor Snaith gave a proof by a radically different approach, topological in nature (an application of the Lefschetz fixed-point theorem). There has been related recent work on the question of finding natural and explicit forms of Brauer's theorem, notably by Robert Boltje.

Characteristically simple group

In mathematics, in the field of group theory, a group is said to be characteristically simple if it has no proper nontrivial characteristic subgroups. Characteristically simple groups are sometimes also termed elementary groups. Characteristically simple is a weaker condition than being a simple group, as simple groups must not have any proper nontrivial normal subgroups, which include characteristic subgroups.

A finite group is characteristically simple if and only if it is the direct product of isomorphic simple groups. In particular, a finite solvable group is characteristically simple if and only if it is an elementary abelian group. This does not hold in general for infinite groups; for example, the rational numbers form a characteristically simple group that is not a direct product of simple groups.

A minimal normal subgroup of a group G is a nontrivial normal subgroup N of G such that the only proper subgroup of N that is normal in G is the trivial subgroup. Every minimal normal subgroup of a group is characteristically simple. This follows from the fact that a characteristic subgroup of a normal subgroup is normal.

Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, G/N is abelian if and only if N contains the commutator subgroup of G. So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is.

Core (group theory)

In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the p-core of a group.

Fitting length

In mathematics, especially in the area of algebra known as group theory, the Fitting length (or nilpotent length) measures how far a solvable group is from being nilpotent. The concept is named after Hans Fitting, due to his investigations of nilpotent normal subgroups.

Hall subgroup

In mathematics, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They were introduced by the group theorist Philip Hall (1928).

Hall–Higman theorem

In mathematical group theory, the Hall–Higman theorem, due to Philip Hall and Graham Higman (1956, Theorem B), describes the possibilities for the minimal polynomial of an element of prime power order for a representation of a p-solvable group.

Imperfect group

In mathematics, in the area of algebra known as group theory, an imperfect group is a group with no nontrivial perfect quotients. Some of their basic properties were established in (Berrick & Robinson 1993). The study of imperfect groups apparently began in (Robinson 1972).The class of imperfect groups is closed under extension and quotient groups, but not under subgroups. If G is a group, N, M are normal subgroups with G/N and G/M imperfect, then G/(N∩M) is imperfect, showing that the class of imperfect groups is a formation. The (restricted or unrestricted) direct product of imperfect groups is imperfect.

Every solvable group is imperfect. Finite symmetric groups are also imperfect. The general linear groups PGL(2,q) are imperfect for q an odd prime power. For any group H, the wreath product H wr Sym2 of H with the symmetric group on two points is imperfect. In particular, every group can be embedded as a two-step subnormal subgroup of an imperfect group of roughly the same cardinality (2|H|2).

Metanilpotent group

In mathematics, in the field of group theory, a metanilpotent group is a group that is nilpotent by nilpotent. In other words, it has a normal nilpotent subgroup such that the quotient group is also nilpotent.

In symbols, is metanilpotent if there is a normal subgroup such that both and are nilpotent.

The following are clear:

Minimal counterexample

In mathematics, the method of considering a minimal counterexample combines the ideas of inductive proof and proof by contradiction. Abstractly, in trying to prove a proposition P, one assumes that it is false, and that therefore there is at least one counterexample. With respect to some idea of size, which may need to be chosen skillfully, one assumes that there is such a counterexample C that is minimal. We expect that C is something quite hypothetical (since we are trying to prove P), but it may be possible to argue that if C existed, it would have some definite properties. From those we then try to get a contradiction.

If the form of the contradiction is that we can derive a further counterexample D, and that D is smaller than C in the sense of the working hypothesis of minimality, then this technique is traditionally called infinite descent. There may however be more complicated ways to argue. For example, the minimal counterexample method has been much used in the classification of finite simple groups. The Feit–Thompson theorem, that finite simple groups that are not cyclic groups have even order, was based on the hypothesis of some, and therefore some minimal, simple group G of odd order. Every proper subgroup of G can be assumed a solvable group, meaning that much theory of such subgroups could be applied.

The assumption that if there is a counterexample, there is a minimal counterexample, is based on a well-ordering of some kind. The usual ordering on the natural numbers is clearly possible, by the most usual formulation of mathematical induction; but the scope of the method is well-ordered induction of any kind.

Euclid's proof of the fundamental theorem of arithmetic is a simple proof using a minimal counterexample.

Monomial group

In mathematics, in the area of algebra studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters are all monomial, that is, induced from characters of degree 1 (Isaacs 1994).

In this section only finite groups are considered. A monomial group is solvable by (Taketa 1930), presented in textbook in (Isaacs 1995, Cor. 5.13) and (Bray et al. 1982, Cor 2.3.4). Every supersolvable group (Bray et al. 1982, Cor 2.3.5) and every solvable A-group (Bray et al. 1982, Thm 2.3.10) is a monomial group. Factor groups of monomial groups are monomial, but subgroups need not be, since every finite solvable group can be embedded in a monomial group, as shown by (Dade & ????) and in textbook form in (Bray et al. 1982, Ch 2.4).

The Symmetric group is an example of a monomial group which is neither supersolvable nor a A-group.

Perfect core

In mathematics, in the field of group theory, the perfect core (or perfect radical) of a group is its largest perfect subgroup. Its existence is guaranteed by the fact that the subgroup generated by a family of perfect subgroups is again a perfect subgroup. The perfect core is also the point where the transfinite derived series stabilizes for any group.

A group whose perfect core is trivial is termed a hypoabelian group. Every solvable group is hypoabelian, and so is every free group. More generally, every residually solvable group is hypoabelian.

The quotient of a group G by its perfect core is hypoabelian, and is called the hypoabelianization of G.

Polycyclic group

In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, and this makes them interesting from a computational point of view.

Primitive permutation group

In mathematics, a permutation group G acting on a non-empty set X is called primitive if G acts transitively on X and G preserves no nontrivial partition of X, where nontrivial partition means a partition that isn't a partition into singleton sets or partition into one set X. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive.

While primitive permutation groups are transitive by definition, not all transitive permutation groups are primitive. The requirement that a primitive group be transitive is necessary only when X is a 2-element set and the action is trivial; otherwise, the condition that G preserves no nontrivial partition implies that G is transitive. This is because for non-transitive actions either the orbits of G form a nontrivial partition preserved by G, or the group action is trivial, in which case any nontrivial partition of X (which exists for |X|≥3) is preserved by G.

This terminology was introduced by Évariste Galois in his last letter, in which he used the French term équation primitive for an equation whose Galois group is primitive.In the same letter he stated also the following theorem.

If G is a primitive solvable group acting on a finite set X, then the order of X is a power of a prime number p, X may be identified with an affine space over the finite field with p elements and G acts on X as a subgroup of the affine group.

An imprimitive permutation group is an example of an induced representation; examples include coset representations G/H in cases where H is not a maximal subgroup. When H is maximal, the coset representation is primitive.

If the set X is finite, its cardinality is called the "degree" of G.

The numbers of primitive groups of small degree were stated by Robert Carmichael in 1937:

Note the large number of primitive groups of degree 16. As Carmichael notes, all of these groups, except for the symmetric and alternating group, are subgroups of the affine group on the 4-dimensional space over the 2-element finite field.

Prosolvable group

In mathematics, more precisely in algebra, a prosolvable group (less common: prosoluble group) is a group that is isomorphic to the inverse limit of an inverse system of solvable groups. Equivalently, a group is called prosolvable, if, viewed as a topological group, every open neighborhood of the identity contains a normal subgroup whose corresponding quotient group is a solvable group.

Residual property (mathematics)

In the mathematical field of group theory, a group is residually X (where X is some property of groups) if it "can be recovered from groups with property X".

Formally, a group G is residually X if for every non-trivial element g there is a homomorphism h from G to a group with property X such that .

More categorically, a group is residually X if it embeds into its pro-X completion (see profinite group, pro-p group), that is, the inverse limit of the inverse system consisting of all morphisms from G to some group H with property X.

Shafarevich's theorem on solvable Galois groups

In mathematics, Shafarevich's theorem states that any finite solvable group is the Galois group of some finite extension of the rational numbers. It was first proved by Igor Shafarevich (1954), though Schmidt later pointed out a gap in the proof, which was fixed by Shafarevich (1989).

Solvable

In mathematics, solvable may refer to:

Solvable group, a group that can be constructed by compositions of abelian groups, or equivalently a group whose derived series reaches the trivial group in finitely many steps

Solvable extension, a field extension whose Galois group is a solvable group

Solvable equation, a polynomial equation whose Galois group is solvable, or equivalently, one whose solutions may be expressed by nested radicals

Solvable Lie algebra, a Lie algebra whose derived series reaches the zero algebra in finitely many steps

Solvable problem, a computational problem that can be solved by a Turing machine

Exactly solvable model in statistical mechanics, a system whose solution can be expressed in closed form, or alternatively, another name for completely integrable systems

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