UNIVERSITY OF MALTA UNIVERSITY OF MALTA

FACULTY OF SCIENCE

Department of Mathematics

B.Sc. 2nd Year

May Session 2000


MA112 Groups (1.5 credits) 24 May 2000

0900-1100


Answer THREE questions

1. Let G be a finite group acting on a finite set X. For x Î X let G(x) and Gx denote, respectively, the orbit and the stabiliser of x. Prove that
|G| = |G(x)|·|Gx|.

Now, suppose that the action is transitive, that is, all the elements of X are in one orbit. For any x Î X let m(x) be the number of permutations under this action which do not fix x. Show that


å
 x Î X 
 m(x) = |G|·(|X|-1).
[Hint. The permutations which do not fix x are those in G-Gx.]



2. Let G be a finite group acting on a finite set X. Let F(g) be the set of elements of X which are fixed by the permutation [^(g)] of X corresponding to g Î G under this action. Prove that the number of orbits of X under this action equals

1
|G|
å
 g Î G 
 |F(g)|.

The faces of a cube are to be painted and c different colours are available. Show that the number of inequivalent cubes which can be obtained this way is

1
24
 (c6+3c4 + 12c3 +8c2).



3. Let G be a finite group and H a subgroup of G of index m. Show that there is an action of G on the left cosets of H whose kernel K is contained in H.

(In the sequel you may assume that the alternating group A5 is simple.)
Let H be a proper subgroup of A5, and suppose its index in A5 is m. Show that m ³ 5 and hence that A5 cannot have subgroups of orders 15, 20 or 30.



4. State carefully the three theorems of Sylow.

Show that a group of order 30 cannot be simple.

File translated from TEX by TTH, version 2.00.
On 29 Sep 2000, 12:12.