Page 103 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.

mož Moravec: Some Topics in the Theory of Finite Groups 91

gap> p1 := (4,5,6)*(7,8,9)*(10,11,12);;
gap> p2 := (4,7,10)*(5,8,11)*(6,9,12);;
gap> p3 := (5,7,6,10)*(8,9,12,11);;
gap> p4 := (5,8,6,12)*(7,11,10,9);;
gap> p5 := (1,4)*(7,8)*(9,11)*(10,12);;
gap> p6 := (1,2)*(7,10)*(8,11)*(9,12);;
gap> p7 := (2,3)*(7,12)*(8,10)*(9,11);;
gap> m11 := Group(p1, p2, p3, p4, p5, p6);;
gap> IsSimple(m11);
true
gap> StructureDescription(m11);
"M11"
gap> m12 := Group(p1, p2, p3, p4, p5, p6,p7);;
gap> IsSimple(m12);
true
gap> StructureDescription(m12);
"M12"

There is a vast amount of properties of finite simple groups that follow from CFSG,
too many to state here. Some of them are:

Theorem 3.3.21 Let S be a finite non-abelian simple group.

1. S can be generated by two elements.

2. Out(S) is a solvable group (used to be Schreier’s conjecture).

3. Every element of S is a commutator (used to be Ore’s conjecture).

CFSG also implies, that, given a positive integer n, there are at most two non-isomorphic
finite simple groups of order n. It may happen that there are two non-isomorphic finite
simple groups of the same order. For example, consider PSL(3, 4) and PSL(4, 2); they are
both of order 20160, and

gap> G:=PSL(4,2);;
gap> H:=PSL(3,4);;
gap> IsomorphismGroups(G,H);
fail

Apart from using GAP, several useful information on finite simple groups can be ob-

tained from Atlas of Finite Group Representations [1].

3.3.5 Problems

1. Complete the proof of Proposition 3.3.1.

2. Prove Proposition 3.3.2.

3. Let G act transitively on X . Suppose that the stabilizer of x ∈ X acts transitively on
X − {x }. Then G acts doubly transitively on X .