Take Home EXAM

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.