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.
• (a) Show that f is a homomorphism.
• (b) If gcd(m, n) = 1, show that f is an isomorphism.
• (c) Find the inverse of f (i.e., find the isomorphism h such that hf = identity and fh = identity) in the case that m = 9 and n = 11.
4. Determine the number of subgroups of Z16 x Z17. Justify your answer.
5. Let N and M be subgroups of a group G.
• (a) If N or M is a normal subgroup of G, show that MN is a subgroup.
• (b) If N and M are normal subgroups of G, show that MN is a normal subgroup of G.