Finite easy group

Finite simple groups are considered to be the building blocks of finite groups in group theory (a branch of mathematics ) .

The finite simple groups play a similar role for the finite groups as the prime numbers do for the natural numbers : Each finite group can be "divided" into its simple groups (for the kind of uniqueness see Jordan-Hölder's theorem ). The reconstruction of a finite group from these “factors” is not unambiguous. However, there are no “still simpler groups” from which the finite simple groups can be constructed.

definition

A group is called simple if it only has and itself as the normal divisor . Here the neutral element of the group is called. Often there is additional demand. ${\displaystyle G}$ ${\displaystyle \left\{e\right\}}$ ${\displaystyle e}$ ${\displaystyle G\neq \left\{e\right\}}$

Since the normal divisors of a group are precisely those subgroups that appear as the core of a group homomorphism , a group is simple if and only if every homomorphic image is isomorphic to or to . Another equivalent definition is: A group is simple if and only if the operation of the group on itself as a group by conjugation is irreducible (that is, the only subgroups invariant under this operation are and ). ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle \left\{e\right\}}$ ${\displaystyle \left\{e\right\}}$ ${\displaystyle G}$

classification

It has been known since 1962 that all non-Abelian finite simple groups must have an even order, because Feit-Thompson's theorem says that groups of odd order are even solvable . However, it was still a long way to the complete classification of the finite simple groups, that is to say, to the enumeration of all finite simple groups except for isomorphism.

In the early 1980s, the leaders of the classification program announced a preliminary conclusion, but larger gaps had to be closed afterwards and not all steps had been published. A new program was launched to simplify the classification and to document it completely. The finite simple groups can be divided into

• cyclic groups of prime order,
• alternating groups with , ${\displaystyle A_{n}}$ ${\displaystyle n>4}$
• Lie-type groups over a finite field (16 infinite families each),
• 26 sporadic groups

To prove the classification theorem

The derivation of the sentence was one of the most extensive projects in the history of mathematics:

• The evidence is spread across over 500 specialist articles with a total of almost 15,000 printed pages. However, not all of the evidence has been published.
• Over 100 mathematicians were involved from the late 1920s to the early 1980s.

However, since parts of the proof could only be done with the help of computers, it is not recognized by all mathematicians. After the “completion” of the proof around 1980, leading mathematicians in the classification program such as Michael Aschbacher and Daniel Gorenstein started a program to simplify the proof and document it completely. Gaps have also been discovered, most of which have been closed without major complications. However, one loophole turned out to be so persistent that Aschbacher and others were only able to provide evidence in 2002 that was 1200 pages long - one reason, however, was that the authors tried to avoid references as much as possible.

Derek John Scott Robinson was more cautious in his 1996 textbook on group theory. He wrote that the given classification is generally believed to be complete, but complete evidence has not yet been recorded.

Ronald Solomon , Richard Lyons and Daniel Gorenstein began a 12-volume presentation of the proof (GLS project) in 1994, which is published by the American Mathematical Society and is expected to be completed in 2023.

Families of finite simple groups

The 16 families of groups of the Lie type together with the cyclic groups of prime order and the alternating groups result in the 18 (infinite) families of the classification theorem.

Cyclic groups with prime order

The cyclic groups with form a family of simple groups. ${\displaystyle C_{p}}$ ${\displaystyle p=2,3,5,7,11,\dots }$

In the case of the finite simple groups, the properties are cyclic and commutative , because every cyclic group is commutative and every finite simple commutative group is cyclic.

For the finite simple groups, the properties of cyclic and odd order almost coincide:

• Every finite simple cyclic group except has an odd number of elements. ${\displaystyle C_{2}}$
• Every finite simple group of odd order is cyclic.

Alternating groups of permutations

The alternating permutation groups with form a family of simple groups. The group A ₅ has 60 elements and is the smallest nonabelian simple group. ${\displaystyle A_{n}}$ ${\displaystyle n>4}$

Lie-type groups

Based on the classification of simple complex Lie algebras , 16 families of simple groups can be constructed, which are named after the corresponding types of Lie algebras. These are

${\displaystyle A_{n}(q)}$, , , , , , , , , , , , , , , . ${\displaystyle B_{n}(q)}$ ${\displaystyle C_{n}(q)}$ ${\displaystyle D_{n}(q)}$ ${\displaystyle E_{6}(q)}$ ${\displaystyle E_{7}(q)}$ ${\displaystyle E_{8}(q)}$ ${\displaystyle F_{4}(q)}$ ${\displaystyle G_{2}(q)}$
${\displaystyle {}^{2}A_{n}(q^{2})}$ ${\displaystyle {}^{2}B_{2}(2^{2m+1})}$ ${\displaystyle {}^{2}D_{n}(q^{2})}$ ${\displaystyle {}^{3}D_{4}(q^{3})}$ ${\displaystyle {}^{2}E_{6}(q^{2})}$ ${\displaystyle {}^{2}F_{4}(2^{2m+1})}$ ${\displaystyle {}^{2}G_{2}(3^{2m+1})}$

For details, see Lie-type group . The groups correspond to the special projective linear groups that are simple with the exception of and . ${\displaystyle A_{n}(q)}$ ${\displaystyle PSL_{n}(q)}$ ${\displaystyle PSL_{2}(2)}$ ${\displaystyle PSL_{2}(3)}$

Since the members of the family are non-Abelian and simple, they agree with their commutator groups . For m = 0 the group is also a member of this infinite family. It's called the tits group and it's simple. Strictly speaking, however, it is not of the Lie type. If one considers the family as an independent infinite family, then although its members are no longer all of the Lie type due to the (only) exception Tits group, it is unnecessary to assign the Tits group a special role elsewhere. ${\displaystyle {}^{2}F_{4}(2^{2m+1})}$ ${\displaystyle {}^{2}F_{4}(2^{2m+1})'}$ ${\displaystyle {}^{2}F_{4}(2)'}$ ${\displaystyle {}^{2}F_{4}(2^{2m+1})'}$

Sporadic groups

A finite simple group is called sporadic if it does not belong to a family with an infinite number of members.

The first 5 of the total of 26 sporadic groups (see there for a tabular overview) were discovered by Émile Mathieu in the years 1862 and 1873.

The 21 “younger” groups were found from 1964 onwards; Most of the time the discovery took place in the context of the search for evidence for the classification theorem. Since some of these groups are quite large, several years passed between their group-theoretical discovery and practical proof of their existence. The largest of all 26 sporadic groups, the so-called monster group with around  elements, was discovered in 1973 by Bernd Fischer and Robert Griess , but Griess did not succeed in its final construction until 1980. ${\displaystyle F_{1}}$ ${\displaystyle 8\times 10^{53}}$

Some authors also count the Tits group with elements among the sporadic groups, which would result in a total of 27. However, it belongs (with ) to the infinite family of -groups of commutator groups of -groups (which, as simple non-Abelian groups, agree with their commutator group ) and is therefore not to be regarded as sporadic. ${\displaystyle {}^{2}F_{4}(2)'}$ ${\displaystyle 17971200=2^{11}\cdot 3^{3}\cdot 5^{2}\cdot 13}$ ${\displaystyle m=0}$ ${\displaystyle {}^{2}F_{4}(2^{2m+1})'}$ ${\displaystyle {}^{2}F_{4}(2^{2m+1})}$ ${\displaystyle m>0}$

Web links and literature

• Dieter Held: The Classification of Finite Simple Groups (PDF, 131 kB)
• Gerhard Hiß: The sporadic groups (PostScript, 460 kB)
• Classification Theorem of Finite Groups at 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…
  • of the journal Spektrum der Wissenschaft , February 1986.
  • and the anthology Moderne Mathematik , Heidelberg 1996, ISBN 3-8274-0025-2 .
• Daniel Gorenstein Finite Simple Groups , Plenum Press 1982
• Ronald Solomon: A brief history of the classification of finite simple groups . BAMS, Vol. 38, 2001, Issue 3, PDF
• Mark Ronan: Symmetry and the Monster . Oxford University Press 2006 (popular science)
• Bernd Stellmacher , Hans Kurzweil: Theory of finite groups - an introduction . Springer 1998, ISBN 354060331X .

GLS Projekt Daniel Gorenstein, Richard Lyons, Ronald Solomon: The classification of the finite simple groups , AMS, Mathematical Surveys and Monographs, Vol. 40., Volume 1, 1994, Volume 2, 1995, Volume 3, 1997, Volume 4, 1999 (Part II, Chapters 1-4: Uniqueness Theorems), Volume 5, 2002, Volume 6, 2004 (Part IV: The special odd case), Volume 7, 2018 (Part III, Chapters 7-11: The Generic Case, Stages 3b and 4a)

swell

1. ↑ Michael Aschbacher : Finite group theory , Cambridge University Press, 1986, ISBN 0-521-45826-9 , p. 9 ff. (English; table of contents , PDF file, 75.4 kB; Zentralblatt review )
2. ^ Aschbacher, Smith: The classification of quasithin groups , AMS
3. ^ DJS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , page 79: More Simple Groups
4. ↑ Ronald Solomon: The Classification of finite simple groups: A progress report , Notices AMS, June / July 2018, online
5. ↑ Solomon, Notices AMS, June / July 2018, Volume 8 will probably appear in 2018. Other mathematicians such as Gernot Stroth , Richard Foote and Inna Capdeboscq are also involved in the project.
6. ↑ AMS

See also

• List of small groups