Page 102 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 102
3.3 Finite simple groups
Proposition 3.3.20 If |F | > 3, then SL(2, F ) is a perfect group.
PROOF. If |F | > 3 there exists a ∈ F such that a 2 ∈/ {0, 1}. Now we observe
1 b (a 2 − 1) = 1/a 0 , 1 0 .
0 1 0 a −b 1
Letting b run through F , we see that U ≤ SL2(F ) . By Proposition 3.3.19 we conclude the
result.
PROOF.[Proof of Theorem 3.3.17 for n = 2] This follows by previous propositions and Iwa-
sawa’s lemma.
3.3.4 On the classification of finite simple groups (CFSG)
One of the greatest achievements of mathematics is a full classificiation of finite simple
groups (CFSG) which was announced in the 1980’s. Roughly speaking, the result says
that all finite simple groups fall into one of the following four types:
1. Cyclic groups of prime order;
2. Alternating groups An for n ≥ 5;
3. Groups of Lie type; these groups arise as automorphism groups of simple Lie alge-
bras. An example is PSL(n, F ).
4. 26 sporadic groups; these do not fall into any infinite family of simple groups de-
scribed above. They are usually defined as symmetry groups of various algebraic
or combinatorial configurations. The largest of them has order
808, 017, 424, 794, 512, 875, 886, 459, 904, 961, 710, 757, 005, 754, 368, 000, 000, 000
and is called the Monster Group.
Since a thorough account on these groups is beyond the purpose of these notes, we only
exhibit some of their properties and how to use GAP to study them. The following are all
non-abelian finite simple groups of order ≤ 1000000:
gap> AllSmallNonabelianSimpleGroups( [1..1000000] );
[ A5, PSL(2,7), A6, PSL(2,8), PSL(2,11), PSL(2,13), PSL(2,17), A7, PSL(2,19),
PSL(2,16), PSL(3,3), PSU(3,3), PSL(2,23), PSL(2,25), M11, PSL(2,27),
PSL(2,29), PSL(2,31), A8, PSL(3,4), PSL(2,37), PSp(4,3), Sz(8), PSL(2,32),
PSL(2,41), PSL(2,43), PSL(2,47), PSL(2,49), PSU(3,4), PSL(2,53), M12,
PSL(2,59), PSL(2,61), PSU(3,5), PSL(2,67), J_1, PSL(2,71), A9, PSL(2,73),
PSL(2,79), PSL(2,64), PSL(2,81), PSL(2,83), PSL(2,89), PSL(3,5), M22,
PSL(2,97), PSL(2,101), PSL(2,103), J_2, PSL(2,107), PSL(2,109), PSL(2,113),
PSL(2,121), PSL(2,125), PSp(4,4) ]
Here is a construction of Mathieu groups M 11 and M 12 which are sporadic groups:
Proposition 3.3.20 If |F | > 3, then SL(2, F ) is a perfect group.
PROOF. If |F | > 3 there exists a ∈ F such that a 2 ∈/ {0, 1}. Now we observe
1 b (a 2 − 1) = 1/a 0 , 1 0 .
0 1 0 a −b 1
Letting b run through F , we see that U ≤ SL2(F ) . By Proposition 3.3.19 we conclude the
result.
PROOF.[Proof of Theorem 3.3.17 for n = 2] This follows by previous propositions and Iwa-
sawa’s lemma.
3.3.4 On the classification of finite simple groups (CFSG)
One of the greatest achievements of mathematics is a full classificiation of finite simple
groups (CFSG) which was announced in the 1980’s. Roughly speaking, the result says
that all finite simple groups fall into one of the following four types:
1. Cyclic groups of prime order;
2. Alternating groups An for n ≥ 5;
3. Groups of Lie type; these groups arise as automorphism groups of simple Lie alge-
bras. An example is PSL(n, F ).
4. 26 sporadic groups; these do not fall into any infinite family of simple groups de-
scribed above. They are usually defined as symmetry groups of various algebraic
or combinatorial configurations. The largest of them has order
808, 017, 424, 794, 512, 875, 886, 459, 904, 961, 710, 757, 005, 754, 368, 000, 000, 000
and is called the Monster Group.
Since a thorough account on these groups is beyond the purpose of these notes, we only
exhibit some of their properties and how to use GAP to study them. The following are all
non-abelian finite simple groups of order ≤ 1000000:
gap> AllSmallNonabelianSimpleGroups( [1..1000000] );
[ A5, PSL(2,7), A6, PSL(2,8), PSL(2,11), PSL(2,13), PSL(2,17), A7, PSL(2,19),
PSL(2,16), PSL(3,3), PSU(3,3), PSL(2,23), PSL(2,25), M11, PSL(2,27),
PSL(2,29), PSL(2,31), A8, PSL(3,4), PSL(2,37), PSp(4,3), Sz(8), PSL(2,32),
PSL(2,41), PSL(2,43), PSL(2,47), PSL(2,49), PSU(3,4), PSL(2,53), M12,
PSL(2,59), PSL(2,61), PSU(3,5), PSL(2,67), J_1, PSL(2,71), A9, PSL(2,73),
PSL(2,79), PSL(2,64), PSL(2,81), PSL(2,83), PSL(2,89), PSL(3,5), M22,
PSL(2,97), PSL(2,101), PSL(2,103), J_2, PSL(2,107), PSL(2,109), PSL(2,113),
PSL(2,121), PSL(2,125), PSp(4,4) ]
Here is a construction of Mathieu groups M 11 and M 12 which are sporadic groups:
