Take Home EXAM
ANSWER 8 OF THE 9 GIVEN PROBLEMS
- Page 108. 11
- Page 115. 6, 9
- Page 117. 13, 31
- Page 108. 15
- Page 118. 27
- Definition: an Abelian group A is torsion if all elements have finite order.
(That is, if the operation is written additively, for every x in A there is a positive integer n, depending on x, such that nx = 0).
Problem: Let Q be the rational numbers under +. Let Z be the integers under
+. Let Q/Z be the quotient group.
Show a) Q/Z is torsion.
b) For every positive integer n, Q/Z has one and only one subgroup
of order n and that group is cyclic.
- Show that a finite abelian group which is not cyclic contains a subgroup which is isomorphic to the direct product C_p x C_p of two cyclic groups of order p for some prime p.