# 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 ($A_7$) $A_7$