Finite simple groups

The cyclic group ${\displaystyle G=(\mathbb {Z} /3\mathbb {Z} ,+)=\mathbb {Z} _{3}}$ of congruence classes modulo 3 (see modular arithmetic) is simple. If ${\displaystyle H}$ is a subgroup of this group, its order (the number of elements) must be a divisor of the order of ${\displaystyle G}$ which is 3. Since 3 is prime, its only divisors are 1 and 3, so either ${\displaystyle H}$ is ${\displaystyle G}$, or ${\displaystyle H}$ is the trivial group. On the other hand, the group ${\displaystyle G=(\mathbb {Z} /12\mathbb {Z} ,+)=\mathbb {Z} _{12}}$ is not simple. The set ${\displaystyle H}$ 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 of the integers ${\displaystyle (\mathbb {Z} ,+)}$ 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 of prime order. The classification of nonabelian simple groups is far less trivial. The smallest nonabelian simple group is the alternating group ${\displaystyle A_{5}}$ of order 60, and every simple group of order 60 is isomorphic to ${\displaystyle A_{5}}$.^[2] The second smallest nonabelian simple group is the projective special linear group PSL(2,7) of order 168, and every simple group of order 168 is isomorphic to PSL(2,7).^[3][4]

Infinite simple groups

The infinite alternating group ${\displaystyle A_{\infty }}$, 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 ${\displaystyle A_{n}}$ with respect to standard embeddings ${\displaystyle A_{n}\rightarrow A_{n+1}}$. Another family of examples of infinite simple groups is given by ${\displaystyle PSL_{n}(F)}$, where ${\displaystyle F}$ is an infinite field and ${\displaystyle n\geq 2}$.

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 ${\displaystyle T}$ and ${\displaystyle V}$. Finitely presented torsion-free infinite simple groups were constructed by Burger and Mozes.^[6]

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).

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

• ${\displaystyle \mathbb {Z} _{p}}$ – cyclic group of prime order
• ${\displaystyle A_{n}}$ – alternating group for ${\displaystyle n\geq 5}$

  The alternating groups may be considered as groups of Lie type over the field with one element, which unites this family with the next, and thus all families of non-abelian finite simple groups may be considered to be of Lie type.

• One of 16 families of groups of Lie type

  The Tits group is generally considered of this form, though strictly speaking it is not of Lie type, but rather index 2 in a group of Lie type.

• One of 26 exceptions, the sporadic groups, of which 20 are subgroups or subquotients of the monster group and are referred to as the "Happy Family", while the remaining 6 are referred to as pariahs.

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.

Notes

Textbooks

• Knapp, Anthony W. (2006), Basic algebra, Springer, ISBN 978-0-8176-3248-9
• Rotman, Joseph J. (1995), An introduction to the theory of groups, Graduate texts in mathematics, vol. 148, Springer, ISBN 978-0-387-94285-8
• Smith, Geoff; Tabachnikova, Olga (2000), Topics in group theory, Springer undergraduate mathematics series (2 ed.), Springer, ISBN 978-1-85233-235-8

Papers