Page 104 - 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. 104
3.4 Some extension theory
4. Let Ω be the set of 2-element subsets of {1, 2, . . . , n }. Then Sn acts on Ω by {i , j }g =
{i g , j g }.
(a) If n = 2, then the action is not faithful.
(b) If n = 3, then the action is doubly transitive.
(c) If n = 4, then the action is imprimitive.
(d) If n ≥ 5, then the action is primitive, but not doubly transitive.
5. Let G be a group. The group AutG acts naturally on the set G .
(a) If G − {1} is an orbit, prove that G is an elementary abelian p -group.
(b) If AutG acts doubly transitively on G − {1}, show that either G is a 2-group or
|G | = 3.
6. Let G be a group of order 2m , where m is odd and m > 1. Prove that G is not
simple.
7. Let n ≥ 2. Show that the transpositions (1 2), (1 3), ..., (1 n ) generate Sn .
8. Let n ≥ 3. Show that the 3-cycles (1 2 3), (1 2 4), ..., (1 2 n ) generate An .
9. Prove that there are no simple groups of order 312, 616, or 1960.
10. Show that the only simple group of order 60 is A5.
11. Prove that PSL(4, 2) ∼= A8.
12. Prove by hand that PSL(3, 4) has no elements of order 15, so it is not isomorphic to
A8.
13. Show that transvections in SL(2, F ) need not be conjugate.
3.4 Some extension theory
Let N be a normal subgroup of G . Then we say that G is an extension of N by G /N .
A precise definition of group extensions will be given in Section 3.4.1. The importance
of extension theory can be outlined as follows. Let G be a finite group and 1 = G0
G1 G2 · · · Gr = G its composition series. By Jordan-Hölder theorem, the composition
factors Gi +1/Gi are in a sense uniquely determined by G . On the other hand, these are
simple groups, so they are known by CFSG. In order to build all finite groups with a given
sequennce of composition factors, one can proceed as follows. Suppose we already know
what Gi is, and we have a prescribed isomorphism type of the simple group Gi +1/Gi . If
we knew how to build all the extensions (up to certain equivalence) of a given group by
a (simple) group, then we would be able to construct all possible Gi +1. Proceeding this
way, we would eventually be able to construct all finite groups. The trouble is that the
problem of constructing all possible extensions is very difficult and still open.
4. Let Ω be the set of 2-element subsets of {1, 2, . . . , n }. Then Sn acts on Ω by {i , j }g =
{i g , j g }.
(a) If n = 2, then the action is not faithful.
(b) If n = 3, then the action is doubly transitive.
(c) If n = 4, then the action is imprimitive.
(d) If n ≥ 5, then the action is primitive, but not doubly transitive.
5. Let G be a group. The group AutG acts naturally on the set G .
(a) If G − {1} is an orbit, prove that G is an elementary abelian p -group.
(b) If AutG acts doubly transitively on G − {1}, show that either G is a 2-group or
|G | = 3.
6. Let G be a group of order 2m , where m is odd and m > 1. Prove that G is not
simple.
7. Let n ≥ 2. Show that the transpositions (1 2), (1 3), ..., (1 n ) generate Sn .
8. Let n ≥ 3. Show that the 3-cycles (1 2 3), (1 2 4), ..., (1 2 n ) generate An .
9. Prove that there are no simple groups of order 312, 616, or 1960.
10. Show that the only simple group of order 60 is A5.
11. Prove that PSL(4, 2) ∼= A8.
12. Prove by hand that PSL(3, 4) has no elements of order 15, so it is not isomorphic to
A8.
13. Show that transvections in SL(2, F ) need not be conjugate.
3.4 Some extension theory
Let N be a normal subgroup of G . Then we say that G is an extension of N by G /N .
A precise definition of group extensions will be given in Section 3.4.1. The importance
of extension theory can be outlined as follows. Let G be a finite group and 1 = G0
G1 G2 · · · Gr = G its composition series. By Jordan-Hölder theorem, the composition
factors Gi +1/Gi are in a sense uniquely determined by G . On the other hand, these are
simple groups, so they are known by CFSG. In order to build all finite groups with a given
sequennce of composition factors, one can proceed as follows. Suppose we already know
what Gi is, and we have a prescribed isomorphism type of the simple group Gi +1/Gi . If
we knew how to build all the extensions (up to certain equivalence) of a given group by
a (simple) group, then we would be able to construct all possible Gi +1. Proceeding this
way, we would eventually be able to construct all finite groups. The trouble is that the
problem of constructing all possible extensions is very difficult and still open.
