Classification of Simple Finite Groups

Notation: n, l, d, m ∈ ℕ and q = p^{m} with p ∈ ℙ throughout this article.

A finite group G is called simple, iff, except the automorphisms f_{i}: G ↦ G, there is exactly one further homomorphism h of G into G, which trivially must be the map h: G → {e}. If G is additionally non-commutative, then Aut(G) ≅ G.

A natural classification of the finite simple groups needs the aspect of Lie-Algebra theory, at least. There are exactly five main-classes of simple groups:

1. Cylic groups ℤ_{p} p ∈ ℙ These are the only commutative simple groups.

2. Alternating groups A_{n} n ≥ 3 and n ≠ 4

Lie-type Chevalley groups (16 families)

Classical linear groups — long known and interpreted as subgroups of n×n matrices over 𝔽_{q} with determinant = 1 and group-operation ˆ= matrix-multiplication.

– PSL(n, q) projective special linear group, n ≥ 2, except if n = 2 then q ∉ {2, 3}.

– PSp(2l, q) projective sympletic linear group, 2l ≥ 2, except if 2l = 2 then q ∉ {2, 3} and if 2l = 4 then q ≠ 2.

– PSU(n, q^{2}) projective special unitary group, n ≥ 2, except if n = 2 then q ∉ {2, 3} and if n = 3 then q ≠ 2.

– PΩ(2l+1, q) l ≥ 2 and q ≠ 2^{m}

– PΩ^{ε}(2l, q) with ε ∈ {−1, +1} l ≥ 3. Together with ten further exceptional families, they are nicely classified by

the representation theory of Lie-Algebras via Dynkin-diagrams into:

3. Untwisted Lie-type groups — 9 families (4 classical and 5 exceptional)

A_{l}(q) ≅ PSL(l+1, q) l ≥ 1

B_{l}(q) ≅ PΩ(2l+1, q) l ≥ 2

C_{l}(q) ≅ PSp(2l, q) l ≥ 2

D_{l}(q) ≅ PΩ^{+}(2l, q) l ≥ 3

G_{2}(q) | · | = q^{6}(q^{6}−1)(q^{2}−1)

F_{4}(q) | · | = q^{24}(q^{12}−1)(q^{8}−1)(q^{6}−1)(q^{2}−1)

E_{6}(q) | · | = \frac{q^{36}(q^{12}−1)(q^{9}−1)(q^{8}−1)(q^{6}−1)(q^{5}−1)(q^{2}−1)}{gcd(3, q−1)}

E_{7}(q) | · | = \frac{q^{63}(q^{18}−1)(q^{14}−1)(q^{12}−1)(q^{10}−1)(q^{8}−1)(q^{6}−1)(q^{2}−1)}{gcd(2, q−1)}

E_{8}(q) | · | = q^{120}(q^{30}−1)(q^{24}−1)(q^{20}−1)(q^{18}−1)(q^{14}−1)(q^{12}−1)(q^{8}−1)(q^{2}−1)

4. Twisted Lie-type groups — 7 families (2 classical and 2+3 exceptional)

2A_{l}(q) ≅ PSU(l+1, q^{2}) l ≥ 1

2D_{l}(q) ≅ PΩ^{−}(2l, q) l ≥ 2

3D_{4}(q) | · | = q^{12}(q^{8}+q^{4}+1)(q^{6}−1)(q^{2}−1)

2E_{6}(q) | · | = \frac{q^{36}(q^{12}−1)(q^{9}+1)(q^{8}−1)(q^{6}−1)(q^{5}+1)(q^{2}−1)}{gcd(3, q+1)}

2B_{2}(q =2^{2m+1}) ≅ ^{2}C_{2}(q =2^{2m+1}) m ≥ 1 | · | = q^{2}(q^{2}+1)(q−1)

2G_{2}(q =3^{2m+1}) m ≥ 1 | · | = q^{3}(q^{3}+1)(q−1)

2F_{4}(q =2^{2m+1}) m ≥ 1 | · | = q^{12}(q^{6}+1)(q^{4}−1)(q^{3}+1)(q−1)

5. Sporadic Groups (27 cases)

group  Z order  isomorphism
T  2^{11} · 3^{3} · 5^{2} · 13 = 2^{4} ∏_{i=25}^{27} 4i  T ·2 ≅ ^{2}F_{4}(2)
M_{11}  1 2^{4} · 3^{2} · 5 · 11 = ∏_{i=8}^{11} i  M_{11} ≅ Aut(S(4, 5, 11))
M_{12}  2 2^{6} · 3^{3} · 5 · 11 = ∏_{i=8}^{12} i  M_{12} ≅ Aut(S(5, 6, 12))
M_{22}  2 2^{7} · 3^{2} · 5 · 7 · 11 = 16·3 ∏_{i=20}^{22} i  2·M_{22} ≅ Aut(S(3, 6, 22))
M_{23}  1 2^{7} · 3^{2} · 5 · 7 · 11 · 23 = 16·3 ∏_{i=20}^{23} i  M_{23} ≅ Aut(S(4, 7, 23))
M_{24}  1 2^{10} · 3^{3} · 5 · 7 · 11 · 23 = 16·3 ∏_{i=20}^{24} i  M_{24} ≅ Aut(S(5, 8, 24))
J_{2} =F_{5−}  2 2^{7} · 3^{3} · 5^{2} · 7 = ∏_{i=4}^{10} i
Suz =F_{3−} 2 2^{13} · 3^{7} · 5^{2} · 7 · 11 · 13 = 72 ∏_{i=1}^{13} i = 72 · 13!
HS  2 2^{9} · 3^{2} · 5^{3} · 7 · 11 = ∏_{i=7}^{11} 2i  HS ≅ Co_{7}
McL  2 2^{7} · 3^{6} · 5^{3} · 7 · 11 = 45 ∏_{i=1}^{11} i  McL·2 ≅ Co_{5}
Co_{3}  1 2^{10} · 3^{7} · 5^{3} · 7 · 11 · 23 = 2^{4}·3^{5} ∏_{i=20}^{25} i
Co_{2}  1 2^{18} · 3^{6} · 5^{3} · 7 · 11 · 23 = 81 ∏_{i=20}^{25} 4i
Co_{1} =F_{2−} 1 2^{21} · 3^{9} · 5^{4} · 7^{2} · 11 · 13 · 23 = 2^{3} · 3^{4} · 5 ∏_{i=20}^{28} 2i 2·Co_{1} ≅ Aut(Λ_{24})
He=F_{7+} 2 2^{10} · 3^{3} · 5^{2} · 7^{3} · 17 = 2 · 6 · 7 · 8 · 48 · 49 · 50 · 51
F i_{22}  2 2^{17} · 3^{9} · 5^{2} · 7 · 11 · 13 = 12 ∏_{i=10}^{15} 12i
F i_{23}  1 2^{18} · 3^{13} · 5^{2} · 7 · 11 · 13 · 17 · 23
F i′_{24} =F_{3+} 2 2^{21} · 3^{16} · 5^{2} · 7^{3} · 11 · 13 · 17 · 23 · 29
F_{5+} =HN 2 2^{14} · 3^{6} · 5^{6} · 7 · 11 · 19 = 30 ∏_{i=19}^{22} 60i
F_{3|3} =T h 1 2^{15} · 3^{10} · 5^{3} · 7^{2} · 13 · 19 · 31
F_{2+} =B  1 2^{41} · 3^{13} · 5^{6} · 7^{2} · 11 · 13 · 17 · 19 · 23 · 31 · 47 = ∏_{i=33}^{36} 2880i ∏_{i=91}^{96} 2i
F_{1} =M  1 2^{46} · 3^{20} · 5^{9} · 7^{6} · 11^{2} · 13^{3} · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
J_{1}  1 2^{3} · 3 · 5 · 7 · 11 · 19 = ∏_{i=19}^{22} i
J_{3}  2 2^{7} · 3^{5} · 5 · 17 · 19 = 27 ∏_{i=17}^{20} 2i = 36 ∏_{i=15}^{19} i
Ru  1 2^{14} · 3^{3} · 5^{3} · 7 · 13 · 29 = 2 ∏_{i=25}^{29} 2i
O′N  2 2^{9} · 3^{4} · 5 · 7^{3} · 11 · 19 · 31 = 180 ∏_{i=341}^{343} 4i
Ly  1 2^{8} · 3^{7} · 5^{6} · 7 · 11 · 31 · 37 · 67
J_{4}  1 2^{21} · 3^{3} · 5 · 7 · 11^{3} · 23 · 29 · 31 · 37 · 43

group-isomorphisms  order structur
A_{3} ≅ ℤ_{3}  3  simple
PSL(2, 2) ≅ S_{3}  6 nonsimple
PSL(2, 3) ≅ A_{4}  12 nonsimple
2B_{2}(2) ≅ ^{2}C_{2}(2) ≅ F rob(5, 4)  20 nonsimple
PSL(2, 4) ≅ PSL(2, 5) ≅ ^{2}D_{2}(2) ≅ A_{5}  60  simple
2A_{2}(2) ≅ PSU(3, 4) ≅ 3^{2}·Q_{8}  72 nonsimple
PSL(2, 7) ≅ PSL(3, 2)  168  simple
PSL(2, 9) ≅ B_{2}(2)′ ≅ C_{2}(2)′ ≅ A_{6}  360  simple
B_{2}(2) ≅ PΩ(5, 2) ≅ PSp(4, 2) ≅ S_{6}  720 nonsimple
3·A_{1}(8) ≅ 3·PSL(2, 8) ≅ ^{2}G_{2}(3)  1512 nonsimple
PSL(4, 2) = A_{8}  2520  simple
G_{2}(2) ≅ ^{2}A_{2}(3)·2 ≅ PSU(3, 9)·2  12096 nonsimple
2A_{3}(2) ≅ PSU(4, 4) ≅ ^{2}D_{3}(2) ≅ C_{2}(3) ≅ PSp(4, 3) ≅ B_{2}(3)  25920  simple
2F_{4}(2) ≅ T ·2  35942400 nonsimple

PSU(2, q^{2}) ≅ PSL(2, q) ≅ PΩ(3, q) ≅ PSp(2, q) ⇔ ^{2}A_{1}(q) ≅ A_{1}(q) ≅ B_{1}(q) ≅ C_{1}(q)
PΩ(5, q) ≅ PSp(4, q)  ⇔  B_{2}(q) ≅ C_{2}(q)
PΩ^{+}(4, q) ≅ PSL(2, q) × PSL(2, q)  ⇔  D_{2}(q) ≅ A_{1}(q) × A_{1}(q)
PΩ^{+}(6, q) ≅ PSL(4, q)  ⇔  D_{3}(q) ≅ A_{3}(q)
PΩ^{−}(4, q) ≅ PSL(2, q^{2})  ⇔  ^{2}D_{2}(q) ≅ ^{2}A_{1}(q^{2})
PΩ^{−}(6, q) ≅ PSU(4, q^{2})  ⇔  ^{2}D_{3}(q) ≅ ^{2}A_{3}(q)
l ≥ 3 , q odd ⇒  PΩ(2l+1, q) ≅ P̸Sp(2l, q) ⇔  B_{l}(q) ≅ C̸_{l}(q)
l ≥ 3 , q even ⇒  PΩ(2l+1, q) ≅ PSp(2l, q) ⇔  B_{l}(q) ≅ C_{l}(q)

|ℤ_{p}| = p
|A_{n}| = \frac{1}{2}n!  n ≥ 2
|PSL(n, q)| = d^{−1}q^{(n^{2}−n)/2} ∏_{i=2}^{n} q^{i}−1 = d^{−1}q^{n−1} ∏_{i=0}^{n−2} q^{n}−q^{i}  d = gcd(n, q−1)
|PSU(n, q^{2})| = d^{−1}q^{(n^{2}−n)/2} ∏_{i=2}^{n} q^{i}−(−1)^{i}  d = gcd(n, q+1)
|PSp(2l, q)| = d^{−1}q^{l^{2}} ∏_{i=1}^{l} q^{2i}−1  d = gcd(2, q−1)
|PΩ(2l+1, q)| = d^{−1}q^{l^{2}} ∏_{i=1}^{l} q^{2i}−1  d = gcd(2, q−1)
|PΩ^{+}(2l, q)| = d^{−1}q^{l^{2}−l}(q^{l}−1) ∏_{i=1}^{l−1} q^{2i}−1  d = gcd(4, q^{l}− 1)
|PΩ^{−}(2l, q)| = d^{−1}q^{l^{2}−l}(q^{l}+1) ∏_{i=1}^{l−1} q^{2i}−1  d = gcd(4, q^{l}+1)

Lie-type  Out = ℤ_{d} × ℤ_{f} × G
family condition  d  f  G |G|·log_{p} q
A_{1}(q)  n = 1  gcd(2, q−1) q = p^{m}  {e} m
A_{n}(q)  n ≥ 2  gcd(n+1, q−1) q = p^{m}  ℤ_{2} 2m
2A_{n}(q)  n ≥ 2  gcd(n+1, q+1) q = p^{2m}  {e} 2m
B_{2}(q)  n = 2  gcd(2, q−1) q = p^{m}  Z_{2} 2m
2B_{2}(q)  n = 2  1  q = 2^{2m+1} {e} 2m+1
B_{n}(q)  n ≥ 3  gcd(2, q−1) q = p^{m}  {e} m
C_{n}(q)  n ≥ 3  gcd(2, q−1) q = p^{m}  {e} m
D_{4}(q)  n = 4  gcd(2, q−1)^{2} q = p^{m}  S_{3} 3m
3D_{4}(q)  n = 4  1  q = p^{3m}  {e} 3m
D_{n}(q) n ≥ 4, even gcd(2, q−1)^{2} q = p^{m}  ℤ_{2} 2m
D_{n}(q) n ≥ 4, odd gcd(4, q^{n}−1) q = p^{m}  ℤ_{2} 2m
2D_{n}(q)  n ≥ 4  gcd(4, q^{n}+1) q = p^{2m}  {e} 2m
G_{2}(q)  n = 2  1  q = p^{m}  ℤ_{2} 2m
2G_{2}(q)  n = 2  1  q = 3^{2m+1} {e} 2m+1
F_{4}(q)  n = 4  1  q = p^{m}  ℤ_{2} 2m
2F_{4}(q)  n = 4  1  q = 2^{2m+1} {e} 2m+1
E_{6}(q)  n = 6  gcd(3, q−1) q = p^{m}  ℤ_{2} 2m
2E_{6}(q)  n = 6  gcd(3, q+1) q = p^{2m}  {e} 2m
E_{7}(q)  n = 7  gcd(2, q−1) q = p^{m}  {e} m
E_{8}(q)  n = 8  1  q = p^{m}  {e} m

|Co_{0} := Aut(Λ_{24})| = (2^{7}\binom{24}{5}/\binom{8}{5} + 2^{12}\binom{24}{1} + 2^{2}\binom{24}{2}2^{12})/48 · |M_{24}|

Co_{4} ≅ 2^{11}·M_{23}

Inn(G) := G/ℤ_{Z(G)} Out(G) := Aut(G)/Inn(G)

Z(A_{n}) = 2 n ≥ 7  Z(A_{6}) = 2 × 2  Z(A_{5}) = 2  Z(A_{4}) = 1
Aut(A_{n}) = A_{n} n ≥ 5  Aut(A_{4}) = A_{4}  Aut(A_{3}) = ℤ_{2}
Z(S_{n}) = 1 n ≥ 3

Aut(S_{n}) = S_{n} n ≥ 3 and n ≠ 6 Aut(S_{6}) = S_{6} × ℤ_{2}

Z(ℤ_{p}) = p Aut(ℤ_{p}) = ℤ_{p}^{∗} ≅ ℤ_{p−1}