Classification of finite simple groups

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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 p \mathbb{Z}_p or C_p -- p -- No proper nontrivial subgroup implies cyclic of prime order, prime order implies no proper nontrivial subgroup
2 alternating group natural number n A_n -- n!/2 n = 1,2,4 A5 is simple, alternating groups are simple
3 projective special linear group natural number n (degree), prime power q = p^r (field size) PSL(n,q) A_{n-1}(q) \frac{q^{n(n-1)/2}\prod_{i=2}^n(q^r - 1)}{\operatorname{gcd}(n,q - 1)} PSL(2,2) = A_1(2), PSL(2,3) = A_1(3) Projective special linear group is simple
4 Chevalley group of type B odd natural number n \ge 3 (degree), prime power q = p^r \Omega_n(q) B_{(n-1)/2}(q) q^{((n-1)/2)^2} [\prod_{i=1}^{(n-1)/2} (q^{2i} - 1)]/\operatorname{gcd}(2,q - 1) \Omega_3(2) = B_1(2), \Omega_3(3) = B_1(3), \Omega_5(2) = B_2(2). Although B_2(2) is not simple, B_2(2)' is.
5 projective symplectic group even natural number n (degree), prime power q = p^r (field size) PSp(n,q) C_{n/2}(q) q^{(n/2)^2} [\prod_{i=1}^{n/2} (q^{2i} - 1)]/\operatorname{gcd}(2,q - 1) PSp(2,2) = C_1(2), PSp(2,3) = C_1(3) Projective symplectic group is simple
6 Chevalley group of type D even natural number n (degree), prime power q = p^r (field size) \Omega_n^+(q) D_{n/2}(q) \frac{1}{\operatorname{gcd}(4,q - 1)} q^{(n/2)((n/2)-1)}(q^{n/2}-1)\prod_{i=1}^{(n/2)-1}(q^{2i}-1) \Omega_2^+(q) = D_1(q), \Omega_4^+(q) = D_2(q), \Omega_6^+(q) = D_3(q) (so simple for n \ge 8
7 Suzuki group Parameter m, effectively q = 2^{1 + 2m} Sz(q) = Sz(2^{1 + 2m}) {}^2B_2(q) q^2(q^2 + 1)(q - 1) = 2^{2 + 4m}(2^{2 + 4m} + 1)(2^{1 + 2m} - 1) m = 0, so Sz(2)
8 Ree group Parameter m > 0, effectively q = 3^{1 + 2m} Ree(q) = Ree(3^{1 + 2m}) {}^2G_2(q) q^3(q^3 + 1)(q - 1) m = 0, so Ree(3)

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)
A_1(q) \cong B_1(q) \cong C_1(q). In other words, PSL(2,q) \cong \Omega_3(q) \cong PSp(2,q) for all q. We denote the group as A_1(q) or PSL(2,q).
B_2(q) \cong C_2(q) for all q. In other words, \Omega_5(q) \cong PSp(4,q) for all q. We denote the group as B_2(q).
B_n(2^m) \cong C_n(2^m) for all m,n. In other words, \Omega_{2n+1}(2^m) \cong PSp(2n,2^m) for all m. Note that the n = 1 case is already covered in the preceding collision. We denote the group as B_n(2^m).

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: PSL(2,4) = A_1(4) and also PSL(2,5) = A_1(5)
projective special linear group:PSL(3,2): PSL(3,2) = A_2(2) and also PSL(2,7) = A_1(7).
alternating group projective special linear group alternating group:A5: alternating group A_5, also projective special linear group PSL(2,4) = A_1(4) and PSL(2,5) = A_1(5).
alternating group:A6: alternating group A_6, also projective special linear group PSL(2,9) = A_1(9)
alternating group:A8: alternating group A_8, also projective special linear group PSL(4,2) = A_3(2).
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 M_{11} 7920 2^4 \cdot 3^2 \cdot 5 \cdot 11 10
Mathieu group:M12 M_{12} 95040 2^6 \cdot 3^3 \cdot 5 \cdot 11 15
Mathieu group:M22 M_{22} 443520 2^7 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 12
Mathieu group:M23 M_{23} 10200960 2^7 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11 \cdot 23 17
Mathieu group:M24 M_{24} 244823040 2^{10} \cdot 3^3 \cdot 5 \cdot 7 \cdot 11 \cdot 23 26
Janko group:J1 J_1 175560 2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19 15
Janko group:J2 (also called the Hall-Janko group) J_2 or HJ 604800 2^7 \cdot 3^3 \cdot 5^2 \cdot 7 21
Janko group:J3 J_3 50232960 2^7 \cdot 3^5 \cdot 5 \cdot 17 \cdot 19 21
Janko group:J4 J_4 86775571046077562880 2^{21} \cdot 3^3 \cdot 5 \cdot 7 \cdot 11^3 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 43 62
Conway group:Co1 \operatorname{Co}_1 4157776806543360000 2^{21} \cdot 3^9 \cdot 5^4 \cdot 7^2 \cdot 11 \cdot 13 \cdot 23 101
Conway group:Co2 \operatorname{Co}_2 42305421312000 2^{18} \cdot 3^6 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23 60
Conway group:Co3 \operatorname{Co}_3 495766656000 2^{10} \cdot 3^7 \cdot 5^3 \cdot 7 \cdot 11 \cdot 23 42
Fischer group:Fi22 \operatorname{Fi}_{22} 64561751654400 2^{17} \cdot 3^9 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 65
Fischer group:Fi23 \operatorname{Fi}_{23} 4089470473293004800 2^{18} \cdot 3^{13} \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 23 98
derived subgroup of Fischer group:Fi24 F_{24} 1255205709190661721292800 2^{21} \cdot 3^{16} \cdot 5^2 \cdot 7^3 \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 29 183
Higman-Sims group HS 44352000 2^9 \cdot 3^2 \cdot 5^3 \cdot 7 \cdot 11 24
McLaughlin group McL 898128000 2^7 \cdot 3^6 \cdot 5 \cdot 7 \cdot 11 24
Held group He 4030387200 2^{10} \cdot 3^3 \cdot 5^2 \cdot 7^3 \cdot 17 33
Rudvalis group Ru 145926144000 2^{14} \cdot 3^3 \cdot 5^3 \cdot 7 \cdot 13 \cdot 29 36
Suzuki sporadic group Suz 448345497600 2^{13} \cdot 3^7 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 43
O'Nan group ON 460815505920 2^9 \cdot 3^4 \cdot 5 \cdot 7^3 \cdot 11 \cdot 19 \cdot 31 30
Harada-Norton group HN 273030912000000 2^{14} \cdot 3^6 \cdot 5^6  \cdot 7 \cdot 11 \cdot 19 54
Lyons group Ly 51765179004000000 2^8 \cdot 3^7 \cdot 5^6 \cdot 7 \cdot 11 \cdot 31 \cdot 37 \cdot 67 53
Thompson group Th 90745943887872000 2^{15} \cdot 3^{10} \cdot 5^3 \cdot 7^2 \cdot 13 \cdot 19 \cdot 31 48
baby monster group B 4154781481226426191177580544000000 [SHOW MORE] 184
monster group M 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 n = 5), projective special linear group (PSL(2,4), also PSL(2,5)), projective symplectic group (PSp(2,4), PSp(2,5)), Chevalley group of type B (B_1(4),B_1(5)) A_5, A_1(4),A_1(5),B_1(4),B_1(5),C_1(4),C_1(5)
projective special linear group:PSL(3,2) 168 projective special linear group (PSL(3,2), also PSL(2,7)), projective symplectic group (PSp(2,7)), Chevalley group of type B (B_1(7)) A_2(2), A_1(7), B_1(7), C_1(7).
alternating group:A6 360 alternating group (parameter n = 6), projective special linear group (PSL(2,9)), projective symplectic group (PSp(2,9)), Chevalley group of type B (B_1(9)) A_6, A_1(9), B_1(9), C_1(9). Also, B_2(2)'
projective special linear group:PSL(2,8) 504 projective special linear group (PSL(2,8)), Chevalley group of type B (B_1(8)), projective symplectic group (PSp(2,8)) A_1(8), B_1(8), C_1(8)
projective special linear group:PSL(2,11) 660 projective special linear group (PSL(2,11)),Chevalley group of type B (B_1(11)), projective symplectic group (PSp(2,11)) A_1(11), B_1(11), C_1(11)
projective special linear group:PSL(2,13) 1092 projective special linear group (PSL(2,13)),Chevalley group of type B (B_1(13)), projective symplectic group (PSp(2,13)) A_1(13), B_1(13), C_1(13)
projective special linear group:PSL(2,17) 2448 projective special linear group (PSL(2,17)),Chevalley group of type B (B_1(17)), projective symplectic group (PSp(2,17)) A_1(17), B_1(17), C_1(17)
alternating group:A7 2520 alternating group (A_7) A_7

References

Expository article references

Textbook references