Simple group

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In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

Group (mathematics) Algebraic structure with one binary operation

In mathematics, a group is a set equipped with a binary operation that combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, and help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study.

Normal subgroup subgroup invariant under conjugation

In abstract algebra, a normal subgroup is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G is normal in G if and only if gng−1N for all gG and nN. The usual notation for this relation is .

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The complete classification of finite simple groups, completed in 2004, is a major milestone in the history of mathematics.

Classification of finite simple groups theorem

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below. These groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution.

Examples

Finite simple groups

The cyclic group G = Z/3Z of congruence classes modulo 3 (see modular arithmetic) is simple. If H is a subgroup of this group, its order (the number of elements) must be a divisor of the order of G which is 3. Since 3 is prime, its only divisors are 1 and 3, so either H is G, or H is the trivial group. On the other hand, the group G = Z/12Z is not simple. The set H of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an abelian group is normal. Similarly, the additive group Z of integers is not simple; the set of even integers is a non-trivial proper normal subgroup. [1]

Cyclic group mathematical group that can be generated as the set of powers of a single element

In group theory, a branch of abstract algebra, a cyclic group or monogenous group is a group that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as a power of g in multiplicative notation, or as a multiple of g in additive notation. This element g is called a generator of the group.

Modulo operation Computational operation

In computing, the modulo operation finds the remainder after division of one number by another.

Modular arithmetic Computation modulo a fixed integer

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value—the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the cyclic groups of prime order. The classification of nonabelian simple groups is far less trivial. The smallest nonabelian simple group is the alternating group A5 of order 60, and every simple group of order 60 is isomorphic to A5. [2] The second smallest nonabelian simple group is the projective special linear group PSL(2,7) of order 168, and it is possible to prove that every simple group of order 168 is isomorphic to PSL(2,7). [3] [4]

Prime number Integer greater than 1 that has no positive integer divisors other than itself and 1

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

Alternating group group of even-parity permutations

In mathematics, an alternating group is the group of even permutations of a finite set. The alternating group on a set of n elements is called the alternating group of degree n, or the alternating group on n letters and denoted by An or Alt(n).

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. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.

Infinite simple groups

The infinite alternating group, i.e. the group of even finitely supported permutations of the integers, is simple. This group can be written as the increasing union of the finite simple groups with respect to standard embeddings Another family of examples of infinite simple groups is given by where is an infinite field and

It is much more difficult to construct finitely generated infinite simple groups. The first existence result is non-explicit; it is due to Graham Higman and consists of simple quotients of the Higman group. [5] Explicit examples, which turn out to be finitely presented, include the infinite Thompson groups T and V. Finitely presented torsion-free infinite simple groups were constructed by Burger-Mozes. [6]

Graham Higman British mathematician

Graham Higman FRS was a prominent British mathematician known for his contributions to group theory.

In mathematics, the Higman group, introduced by Graham Higman (1951), was the first example of an infinite finitely presented group with no non-trivial finite quotients. The quotient by the maximal proper normal subgroup is a finitely generated infinite simple group. Higman (1974) later found some finitely presented infinite groups Gn,r that are simple if n is even and have a simple subgroup of index 2 if n is odd, one of which is one of the Thompson groups.

In mathematics, the Thompson groups are three groups, commonly denoted , which were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, F is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group.

Classification

There is as yet no known classification for general (infinite) simple groups, and no such classification is expected.

Finite simple groups

The finite simple groups are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the way prime numbers are the basic building blocks of the integers. This is expressed by the Jordan–Hölder theorem which states that any two composition series of a given group have the same length and the same factors, up to permutation and isomorphism. In a huge collaborative effort, the classification of finite simple groups was declared accomplished in 1983 by Daniel Gorenstein, though some problems surfaced (specifically in the classification of quasithin groups, which were plugged in 2004).

Integer Number in {..., –2, –1, 0, 1, 2, ...}

An integer is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5+1/2, and 2 are not.

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. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module M is a finite increasing filtration of M by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of M into its simple constituents.

Up to Mathematical statement of uniqueness, except for an equivalent . structure (equivalence relation)

In mathematics, the phrase up to appears in discussions about the elements of a set, and the conditions under which subsets of those elements may be considered equivalent. The statement "elements a and b of set S are equivalent up to X" means that a and b are equivalent if criterion X is ignored. That is, a and b can be transformed into one another if a transformation corresponding to X is applied.

Briefly, finite simple groups are classified as lying in one of 18 families, or being one of 26 exceptions:

Structure of finite simple groups

The famous theorem of Feit and Thompson states that every group of odd order is solvable. Therefore, every finite simple group has even order unless it is cyclic of prime order.

The Schreier conjecture asserts that the group of outer automorphisms of every finite simple group is solvable. This can be proved using the classification theorem.

History for finite simple groups

There are two threads in the history of finite simple groups – the discovery and construction of specific simple groups and families, which took place from the work of Galois in the 1820s to the construction of the Monster in 1981; and proof that this list was complete, which began in the 19th century, most significantly took place 1955 through 1983 (when victory was initially declared), but was only generally agreed to be finished in 2004. As of 2010, work on improving the proofs and understanding continues; see ( Silvestri 1979 ) for 19th century history of simple groups.

Construction

Simple groups have been studied at least since early Galois theory, where Évariste Galois realized that the fact that the alternating groups on five or more points are simple (and hence not solvable), which he proved in 1831, was the reason that one could not solve the quintic in radicals. Galois also constructed the projective special linear group of a plane over a prime finite field, PSL(2,p), and remarked that they were simple for p not 2 or 3. This is contained in his last letter to Chevalier, [7] and are the next example of finite simple groups. [8]

The next discoveries were by Camille Jordan in 1870. [9] Jordan had found 4 families of simple matrix groups over finite fields of prime order, which are now known as the classical groups.

At about the same time, it was shown that a family of five groups, called the Mathieu groups and first described by Émile Léonard Mathieu in 1861 and 1873, were also simple. Since these five groups were constructed by methods which did not yield infinitely many possibilities, they were called "sporadic" by William Burnside in his 1897 textbook.

Later Jordan's results on classical groups were generalized to arbitrary finite fields by Leonard Dickson, following the classification of complex simple Lie algebras by Wilhelm Killing. Dickson also constructed exception groups of type G2 and E6 as well, but not of types F4, E7, or E8( Wilson 2009 , p. 2). In the 1950s the work on groups of Lie type was continued, with Claude Chevalley giving a uniform construction of the classical groups and the groups of exceptional type in a 1955 paper. This omitted certain known groups (the projective unitary groups), which were obtained by "twisting" the Chevalley construction. The remaining groups of Lie type were produced by Steinberg, Tits, and Herzig (who produced 3D4(q) and 2E6(q)) and by Suzuki and Ree (the Suzuki–Ree groups).

These groups (the groups of Lie type, together with the cyclic groups, alternating groups, and the five exceptional Mathieu groups) were believed to be a complete list, but after a lull of almost a century since the work of Mathieu, in 1964 the first Janko group was discovered, and the remaining 20 sporadic groups were discovered or conjectured in 1965–1975, culminating in 1981, when Robert Griess announced that he had constructed Bernd Fischer's "Monster group". The Monster is the largest sporadic simple group having order of 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000. The Monster has a faithful 196,883-dimensional representation in the 196,884-dimensional Griess algebra, meaning that each element of the Monster can be expressed as a 196,883 by 196,883 matrix.

Classification

The full classification is generally accepted as starting with the Feit–Thompson theorem of 1962/63, largely lasting until 1983, but only being finished in 2004.

Soon after the construction of the Monster in 1981, a proof, totaling more than 10,000 pages, was supplied that group theorists had successfully listed all finite simple groups, with victory declared in 1983 by Daniel Gorenstein. This was premature – some gaps were later discovered, notably in the classification of quasithin groups, which were eventually replaced in 2004 by a 1,300 page classification of quasithin groups, which is now generally accepted as complete.

Tests for nonsimplicity

Sylow's test : Let n be a positive integer that is not prime, and let p be a prime divisor of n. If 1 is the only divisor of n that is equal to 1 modulo p, then there does not exist a simple group of order n.

Proof: If n is a prime-power, then a group of order n has a nontrivial center [10] and, therefore, is not simple. If n is not a prime power, then every Sylow subgroup is proper, and, by Sylow's Third Theorem, we know that the number of Sylow p-subgroups of a group of order n is equal to 1 modulo p and divides n. Since 1 is the only such number, the Sylow p-subgroup is unique, and therefore it is normal. Since it is a proper, non-identity subgroup, the group is not simple.

Burnside: A non-Abelian finite simple group has order divisible by at least three distinct primes. This follows from Burnside's p-q theorem.

See also

Related Research Articles

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In mathematics, a finite field or Galois field is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number.

<i>p</i>-group

In mathematical group theory, given a prime number p, a p-group is a group in which each element has a power of p as its order. That is, for each element g of a p-group, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p.

Symmetric group automorphism group of a set; the group of bijections on a set, whose group operation is function composition

In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group Sn defined over a finite set of n symbols consists of the permutation operations that can be performed on the n symbols. Since there are n! such permutation operations, the order of the symmetric group Sn is n!.

Solvable group group that can be constructed from abelian groups using extensions; a group whose derived series terminates in the trivial subgroup

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.

Nilpotent group

A nilpotent groupG is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with {1}.

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G : H| or [G : H] or (G:H).

Finite group mathematical group based upon a finite number of elements

In abstract algebra, a finite group is a group, of which the underlying set contains a finite number of elements.

Ferdinand Georg Frobenius German mathematician

Ferdinand Georg Frobenius was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous determinantal identities, known as Frobenius–Stickelberger formulae, governing elliptic functions, and for developing the theory of biquadratic forms. He was also the first to introduce the notion of rational approximations of functions, and gave the first full proof for the Cayley–Hamilton theorem. He also lent his name to certain differential-geometric objects in modern mathematical physics, known as Frobenius manifolds.

Order (group theory) cardinality of a group, or where the element a of a group is the smallest positive integer m such that am = e

In group theory, a branch of mathematics, the term order is used in three different senses:

In number theory and algebraic geometry, a modular curveY(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z). The term modular curve can also be used to refer to the compactified modular curvesX(Γ) which are compactifications obtained by adding finitely many points to this quotient. The points of a modular curve parametrize isomorphism classes of elliptic curves, together with some additional structure depending on the group Γ. This interpretation allows one to give a purely algebraic definition of modular curves, without reference to complex numbers, and, moreover, prove that modular curves are defined either over the field Q of rational numbers, or a cyclotomic field. The latter fact and its generalizations are of fundamental importance in number theory.

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).

In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an abelian group of odd order normalizing a non-trivial 2-subgroup of G. When G is a group of Lie type of characteristic 2 type, the width is usually the rank.

In mathematical finite group theory, an N-group is a group all of whose local subgroups are solvable groups. The non-solvable ones were classified by Thompson during his work on finding all the minimal finite simple groups.

References

Notes

  1. Knapp (2006), p. 170
  2. Rotman (1995), p. 226
  3. Rotman (1995), p. 281
  4. Smith & Tabachnikova (2000), p. 144
  5. Higman, Graham (1951), "A finitely generated infinite simple group", Journal of the London Mathematical Society, Second Series, 26 (1): 61–64, doi:10.1112/jlms/s1-26.1.59, ISSN   0024-6107, MR   0038348
  6. Burger, M.; Mozes, S. (2000). "Lattices in product of trees". Publ. Math. IHES. 92: 151–194. doi:10.1007/bf02698916.
  7. Galois, Évariste (1846), "Lettre de Galois à M. Auguste Chevalier", Journal de Mathématiques Pures et Appliquées , XI: 408–415, retrieved 2009-02-04, PSL(2,p) and simplicity discussed on p. 411; exceptional action on 5, 7, or 11 points discussed on pp. 411–412; GL(ν,p) discussed on p. 410
  8. Wilson, Robert (October 31, 2006), "Chapter 1: Introduction", The finite simple groups
  9. Jordan, Camille (1870), Traité des substitutions et des équations algébriques
  10. See the proof in p-group, for instance.

Textbooks

Papers

  • Silvestri, R. (September 1979), "Simple groups of finite order in the nineteenth century", Archive for History of Exact Sciences, 20 (3–4): 313–356, doi:10.1007/BF00327738