Mathematics 581 Problem Set 3 Due Monday, April 22

1. (a) Prove that if in a group G,
(ab)^{2} = a^{2}b^{2 } for all a
and b in G, then G is Abelian. (Hint: Start
by writing the elements in the equation in full without using exponents.)

(b) Prove that if in a group
G, a^{2} = e for all a in G, then
G is Abelian. (Hint: Show that the condition in part (a) is fulfilled.)

2. Let G = (g) be a finite
cyclic group with n elements. Define a mapping f
of **Z _{n}** into G by

f([k]) = g

3. Let G e a group and H be a subgroup of G. Prove that for any elements a and b of G, either Ha = Hb or Ha and Hb are disjoint . (For some a, b, Ha could equal Hb, while for other a, b, the cosets Ha and Hb could be disjoint.)

4. Let G be a group with exactly p^{n}
elements, where p is prime. Recall that the order
of an element g is the least positive integer k such that g^{k}
= e, and that the cyclic subgroup (g) generated by g consists of the distinct
elements {e, g, g^{2}^{, ..., }g^{k
- 1}}. Why must the order of g be a power of p?

5. Let H be a subgroup of a group G. On G define a relation ~ on G by g ~ k if and only if k¢g is in H.

(a) Prove that ~ is an equivalence
relation.

(b) Prove that the equivalence classes
of ~ are the left cosets of H in G.

(c) How would you define an equivalence
relation on G so that the equivalence classes are the right cosets of H
in G?