Classification of finite simple groups
Finite simple groups, hereinafter referred to as " simple groups ", ( a branch of mathematics ) as the building blocks of finite groups are in group theory.
Simple groups play a similar role as the primes for the natural numbers for finite groups: Every finite group can be constructed from simple groups in finitely many steps; every finite group can be split up into their simple groups again. However, there are no " even simpler groups " from which can construct the finite simple groups.
Definition
A group is called simple if it and themselves as having normal subgroup only. It is often also required. Since the normal subgroup of a group are precisely the subgroups which occur as the core of a group homomorphism, a group is simple if and only if every homomorphic image of isomorphic to or too. Another equivalent definition is: A group is simple if and only if the action of the group on itself is irreducible as a group by conjugation (that is, the only invariant under this operation subgroups are and ).
Classification
Since 1962, it was known that any non- abelian simple groups must have an even order (the set of Feit - Thompson says that groups of odd order are solvable even ). Until the complete classification but it was still a long way.
Since 1982, the finite simple groups are completely classified, they can be divided into
- Cyclic groups of prime order,
- Groups of Lie type, 16 each infinite series,
- Alternating groups with and
- 26 sporadic groups.
For the proof of the classification theorem
The derivation of the sentence was one of the largest projects of the history of mathematics:
- The proof is spread over 500 professional articles with a total of almost 15,000 printed pages. But it is not also been published all the evidence.
- More than 100 mathematicians were involved from the late 1920s to the early 1980s in it.
Since parts of the sentence were checked with the aid of computers, the evidence is not accepted by all mathematicians. After the " completion " of the proof to 1980 by leading mathematicians of the classification program as Michael Aschbacher and Daniel Gorenstein, a program has been added to simplify the proof, and fully documented. It also gaps have been discovered, most of which could be closed without major complications. A gap, however, proved to be so stubborn that only in 2002 was provided by Aschbacher and other evidence, which was at least 1200 pages long - but one reason was that the authors tried to manage as possible without references.
Families of simple groups (examples)
Cyclic groups of prime order
The cyclic group Zp p = 2, 3, 5, 7, 11, ... constitute a family of simple groups.
In the simple groups the properties coincide cyclic and commutative, for every cyclic group is commutative and every simple commutative group is cyclic.
- Any simple cyclic group - except Z2 - has an odd number of elements.
- Any simple group of odd order is one of the cyclic groups.
Alternating permutation
The alternating permutation Altn with n greater than 4 form a family of simple groups.
Sporadic groups
The first 5 of the 26 sporadic groups (see below for a summary table ) have already been discovered by Émile Mathieu in the years 1862 and 1873.
The 21 "younger" groups were found in 1964; usually took place the discovery in the context of proof search for the Classification Theorem. As these groups are quite large in some cases, elapsed between their group theoretical discovery and the practical proof of their existence often several years. The largest of all 26 sporadic groups, the so-called monster group F1 with approximately 8 × 1053 elements, was discovered in 1973, junior Bernd Fischer and Robert Griess, but their final design succeeded semolina until 1980.
Some authors also the group 2F4 ( 2) ' with 17.9712 million = 211.33.52.13 elements counted among the sporadic groups, bringing a total of 27 results.
Links and literature
- Gerhard Hiss: The sporadic groups (Postscript, 460 kB)
- Classification Theorem of Finite Groups with Mathworld
- Michael Aschbacher: The status of the classification of the finite simple groups. Notices AMS 2004, PDF, English
- Daniel Gorenstein: The classification of finite simple groups. In ... the Scientific American February 1986.
- And the anthology Modern Mathematics, Heidelberg 1996, ISBN 3-8274-0025-2.
- Daniel Gorenstein, Lyons, Ronald Solomon: The classification of the finite simple groups. 1994 AMS Books Online, English, Vol 2 1996 is here: