Mathematics 581   Problem Set 3   Due Monday, April 22

1. (a) Prove that if in a group G,  (ab)2 = a2b 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, a2 = 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  Zn  into  G  by
f([k]) = gk. Prove that f is well-defined.

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 pn elements, where  p  is prime.   Recall that the order of an element g is the least positive integer k such that gk = e, and that the cyclic subgroup (g) generated by g consists of the distinct elements {e, g, g2, ..., gk - 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?