| Algebraic structure →Group theory Group theory |
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Infinite dimensional Lie group
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Inmathematics, asimple group is a nontrivialgroup whose onlynormal subgroups are thetrivial 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 correspondingquotient group. This process can be repeated, and forfinite groups one eventually arrives at uniquely determined simple groups, by theJordan–Hölder theorem.
The completeclassification of finite simple groups, completed in 2004, is a major milestone in the history of mathematics.
Thecyclic group ofcongruence classesmodulo 3 (seemodular arithmetic) is simple. If is a subgroup of this group, itsorder (the number of elements) must be adivisor of the order of which is 3. Since 3 is prime, its only divisors are 1 and 3, so either is, or is the trivial group. On the other hand, the group is not simple. The set 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 anabelian group is normal. Similarly, the additive group of theintegers is not simple; the set of even integers is a non-trivial proper normal subgroup.[1]
One may use the same kind of reasoning for any abelian group, to deduce that the only simple abelian groups are the cyclic groups ofprime order. The classification of nonabelian simple groups is far less trivial. The smallest nonabelian simple group is thealternating group of order 60, and every simple group of order 60 isisomorphic to.[2] The second smallest nonabelian simple group is the projective special linear groupPSL(2,7) of order 168, and every simple group of order 168 is isomorphic to PSL(2,7).[3][4]
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 constructfinitely generated infinite simple groups. The first existence result is non-explicit; it is due toGraham Higman and consists of simple quotients of theHigman group.[5] Explicit examples, which turn out to be finitely presented, include the infiniteThompson groups and. Finitely presentedtorsion-free infinite simple groups were constructed by Burger and Mozes.[6]
There is as yet no known classification for general (infinite) simple groups, and no such classification is expected. One reason for this is the existence of continuum-manyTarski monster groups for every sufficiently-large prime characteristic, each simple and having only the cyclic group of that characteristic as its subgroups.[7]
Thefinite simple groups are important because in a certain sense they are the "basic building blocks" of all finite groups, somewhat similar to the wayprime numbers are the basic building blocks of theintegers. This is expressed by theJordan–Hölder theorem which states that any twocomposition series of a given group have the same length and the same factors,up topermutation andisomorphism. In a huge collaborative effort, theclassification of finite simple groups was declared accomplished in 1983 byDaniel Gorenstein, though some problems surfaced (specifically in the classification ofquasithin groups, which were plugged in 2004).
Briefly, finite simple groups are classified as lying in one of 18 families, or being one of 26 exceptions:
The famoustheorem ofFeit andThompson states that every group of odd order issolvable. Therefore, every finite simple group has even order unless it is cyclic of prime order.
TheSchreier conjecture asserts that the group ofouter automorphisms of every finite simple group is solvable. This can be proved using theclassification theorem.
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. By 2018, its publication was envisioned as a series of 12monographs,[8] the tenth of which was published in 2023.[9] See (Silvestri 1979) for 19th century history of simple groups.
Simple groups have been studied at least since earlyGalois theory, whereÉvariste Galois realized that the fact that thealternating 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 theprojective special linear group of a plane over a prime finite field,PSL(2,p), and remarked that they were simple forp not 2 or 3. This is contained in his last letter to Chevalier,[10] and are the next example of finite simple groups.[11]
The next discoveries were byCamille Jordan in 1870.[12] Jordan had found 4 families of simple matrix groups overfinite fields of prime order, which are now known as theclassical groups.
At about the same time, it was shown that a family of five groups, called theMathieu groups and first described byÉmile 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" byWilliam Burnside in his 1897 textbook.
Later Jordan's results on classical groups were generalized to arbitrary finite fields byLeonard Dickson, following the classification ofcomplex simple Lie algebras byWilhelm Killing. Dickson also constructed exception groups of type G2 andE6 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, withClaude 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 produced3D4(q) and2E6(q)) and by Suzuki and Ree (theSuzuki–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 firstJanko group was discovered, and the remaining 20 sporadic groups were discovered or conjectured in 1965–1975, culminating in 1981, whenRobert Griess announced that he had constructedBernd 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-dimensionalGriess algebra, meaning that each element of the Monster can be expressed as a 196,883 by 196,883 matrix.
The full classification is generally accepted as beginning with theFeit–Thompson theorem of 1962–1963 and being completed in 2004.
Soon after the construction of the Monster in 1981, a proof, totaling more than 10,000 pages, was supplied in 1983 by Daniel Gorenstein, that claimed to successfullylist all finite simple groups. This was premature, as gaps were later discovered in the classification ofquasithin groups. The gaps were filled in 2004 by a 1300 page classification of quasithin groups and the proof is now generally accepted as complete.
Sylow's test: Letn be a positive integer that is not prime, and letp be a prime divisor ofn. If 1 is the only divisor ofn that is congruent to 1 modulop, then there does not exist a simple group of ordern.
Proof: Ifn is a prime-power, then a group of ordern has a nontrivialcenter[13] and, therefore, is not simple. Ifn is not a prime power, then every Sylow subgroup is proper, and, bySylow's Third Theorem, we know that the number of Sylowp-subgroups of a group of ordern is equal to 1 modulop and dividesn. Since 1 is the only such number, the Sylowp-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 fromBurnside's theorem.
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