Classification of finite simple groups
The Classification of finite simple groups is a mega-theorem which states that every finite simple group belongs to one of eighteen infinite families of simple groups, or to one of 26 sporadic simple groups.
The eighteen families
Here are the families, up to isomorphism. Note that these families are one-parameter, two-parameter or three-parameter families. Each parameter varies either over prime numbers or over natural numbers. Many of the families have a few small exceptions that turn out not to be simple groups.
Also, some of these families have intersections, i.e., there are some groups that occur in multiple families. The intersection of any two families is finite: there are only finitely many groups that are simultaneously in two distinct families.
|No.||Family name||Nature of parameters||Notation for group||Chevalley notation (if applicable)||Order||Exceptions(not simple)||Links to proofs|
|1||cyclic groups of prime order||prime number||or||--||--||No proper nontrivial subgroup implies cyclic of prime order, prime order implies no proper nontrivial subgroup|
|2||alternating group||natural number||--||A5 is simple, alternating groups are simple|
|3||projective special linear group||natural number (degree), prime power (field size)||Projective special linear group is simple|
|4||Chevalley group of type B||odd natural number (degree), prime power||, , . Although is not simple, is.|
|5||projective symplectic group||even natural number (degree), prime power (field size)||,||Projective symplectic group is simple|
|6||Chevalley group of type D||even natural number (degree), prime power (field size)||, , (so simple for|
|7||Suzuki group||Parameter , effectively||, so|
|8||Ree group||Parameter , effectively||, so|
10 more families need to be entered in the table above.
Collisions between families
Here are some of the infinite collisions:
|Collision||Precedence convention (if any)|
|. In other words, for all .||We denote the group as or .|
|for all . In other words, for all .||We denote the group as .|
|for all . In other words, for all . Note that the case is already covered in the preceding collision.||We denote the group as .|
Here is the list of finite and isolated collisions by family pairs:
|First family||Second family||All the collision cases||Proof|
|projective special linear group||projective special linear group|| alternating group:A5: and also
projective special linear group:PSL(3,2): and also .
|alternating group||projective special linear group|| alternating group:A5: alternating group , also projective special linear group and .
alternating group:A6: alternating group , also projective special linear group
alternating group:A8: alternating group , also projective special linear group .
|projective special linear group equals alternating group in only finitely many cases|
The table needs to be completed.
The twenty-six sporadic simple groups
|Group name||Symbol||Order||Prime factorization of order||Number of conjugacy classes|
|Janko group:J2 (also called the Hall-Janko group)||or||604800||21|
|derived subgroup of Fischer group:Fi24||1255205709190661721292800||183|
|Suzuki sporadic group||448345497600||43|
|baby monster group||4154781481226426191177580544000000||[SHOW MORE]||184|
|monster group||808017424794512875886459904961710757005754368000000000||[SHOW MORE]||194|
List of simple non-abelian groups of small order
Further information: List of simple non-abelian groups of small order
The simple abelian groups are precisely the groups of prime order, and there is one such group for each prime number.
The first few simple non-abelian groups are listed below:
Expository article references
- A brief history of the classification of the finite simple groups by Ronald Mark Solomon, Bulletin of the American Mathematical Society, ISSN 10889485 (electronic), ISSN 02730979 (print), Volume 38,Number 3, Page 315 - 352(Year 2001): An expository paper by Ronald Mark Solomon describing the 110-year history of the classification of finite simple groups.Weblink (PDF)More info
- Paper:AschbacherCFSGMore info