M417 Homework 8 Spring 2004
Instructions: Solutions are due Fri., April 9.
  1. The 3rd Isomorphism Theorem (AKA the Freshman's Delight): Consider groups A < B < C, where A and B are normal subgroups of C. Show that B/A is a normal subgroup of C/A, and that (C/A)/(B/A) is isomorphic to C/B. [Hint: apply Homework problem #7.2 for the first part, and use the 1st isomorphism theorem for the second.]
  2. Find all solutions x to:
               x mod 37 = 17
               x mod 29 =  6
    
  3. Define f : Zm x Zn -> Zmn by f((x, y)) = nx + my mod mn.
  4. Determine the number of subgroups of Z16 x Z17. Justify your answer.
  5. Let N and M be subgroups of a group G.