UNIVERSITY OF MALTA UNIVERSITY OF MALTA

FACULTY OF SCIENCE

Department of Mathematics

B.Sc. Part II

January Session 1995


MA211 Group Theory 31 January 1995

1200-1315


Answer TWO questions

1. Let G be a finite group and H a subgroup of G. Let [G:H] = m and let X be the set of left cosets of H in G. Prove that there is an action of G on H whose kernel K is the largest normal subgroup of G contained in K. Show also that |G|/|K| divides m!.

Now suppose m is prime, and it is the smallest prime divisor of |G|. Prove that H is a normal subgroup of G.


[If required, the First Isomorphism Theorem may be quoted without proof.]



2. Let G be a finite group. Write down the class equation for G, explaining clearly all the terms involved and the associated group action.

In the sequel let G be a p-group. Prove that the centre Z(G) of G is nontrivial. Show that if G acts on a set X, and X1 is the set of fixed points of this action, then

|X1| = |X| mod p.
Deduce that if H is a normal subgroup of G then |HÇZ(G)| ³ p.


[The Orbit-Stabiliser Theorem may be quoted without proof.]



3. State Sylow's Theorems.

Show that a group of order 56 cannot be simple.

Let G be a group of order 110. How many elements of order 11 are there in G? What are the possible numbers of elements of order 5 in G? Give two examples of groups of order 110 having different numbers of elements of order 2.

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