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, a normal subgroup and the quotient group, and the process can be repeated. If the group is finite, then eventually one arrives at uniquely determined simple groups by the Jordan–Hölder theorem. The complete classification of finite simple groups, completed in 2008, is a major milestone in the history of mathematics.


Finite simple groups


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]

Infinite simple groups

The infinite alternating group, i.e. the group of even permutations of the integers, is simple. This group can be defined 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 a field and .

It is much more difficult to construct finitely generated infinite simple groups. The first example is due to Graham Higman and is a quotient of the Higman group.[5] Other examples include the infinite Thompson groups T and V. Finitely presented torsion-free infinite simple groups were constructed by Burger-Mozes.[6]


There is as yet no known classification for general simple groups.

Finite simple groups



Structure of finite simple groups



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. Template:As of, work on improving the proofs and understanding continues; see Template:Harv for 19th century history of simple groups.


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.



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.


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




See also



  1. Knapp (2006), [[[:Template:Google books]] p. 170]
  2. Rotman (1995), [[[:Template:Google books]] p. 226]
  3. Rotman (1995), p. 281
  4. Smith & Tabachnikova (2000), [[[:Template:Google books]] p. 144]
  6. M. Burger and S. Mozes. " Lattices in product of trees." Publ. Math. IHES 92 (2000), pp.151–194.
  10. See the proof in p-group, for instance.



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External links