Classification of finite simple groups

From Groupprops

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
Mathieu group:M11 7920 10
Mathieu group:M12 95040 15
Mathieu group:M22 443520 12
Mathieu group:M23 10200960 17
Mathieu group:M24 244823040 26
Janko group:J1 175560 15
Janko group:J2 (also called the Hall-Janko group) or 604800 21
Janko group:J3 50232960 21
Janko group:J4 86775571046077562880 62
Conway group:Co1 4157776806543360000 101
Conway group:Co2 42305421312000 60
Conway group:Co3 495766656000 42
Fischer group:Fi22 64561751654400 65
Fischer group:Fi23 4089470473293004800 98
derived subgroup of Fischer group:Fi24 1255205709190661721292800 183
Higman-Sims group 44352000 24
McLaughlin group 898128000 24
Held group 4030387200 33
Rudvalis group 145926144000 36
Suzuki sporadic group 448345497600 43
O'Nan group 460815505920 30
Harada-Norton group 273030912000000 54
Lyons group 51765179004000000 53
Thompson group 90745943887872000 48
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:

Group Order Families of simple non-abelian groups that it is a member of Shorthand notations
alternating group:A5 60 alternating group (parameter ), projective special linear group (, also ), projective symplectic group (), Chevalley group of type B ()
projective special linear group:PSL(3,2) 168 projective special linear group (, also ), projective symplectic group (), Chevalley group of type B () .
alternating group:A6 360 alternating group (parameter ), projective special linear group (), projective symplectic group (), Chevalley group of type B () . Also,
projective special linear group:PSL(2,8) 504 projective special linear group (), Chevalley group of type B (), projective symplectic group ()
projective special linear group:PSL(2,11) 660 projective special linear group (),Chevalley group of type B (), projective symplectic group ()
projective special linear group:PSL(2,13) 1092 projective special linear group (),Chevalley group of type B (), projective symplectic group ()
projective special linear group:PSL(2,17) 2448 projective special linear group (),Chevalley group of type B (), projective symplectic group ()
alternating group:A7 2520 alternating group ()

References

Expository article references

Textbook references