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 $p$	$\mathbb{Z}_p$ 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] $2^{41} \cdot 3^{13} \cdot 5^6 \cdot 7^2 \cdot 11 \cdot 13\cdot 17\cdot 19 \cdot 23 \cdot 31 \cdot 47$	184
monster group	$M$	808017424794512875886459904961710757005754368000000000	[SHOW MORE] $2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71$	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

• 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:AschbacherCFSG^{More info}