- The 3
^{rd}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 1^{st}isomorphism theorem for the second.] - Find all solutions x to:
x mod 37 = 17 x mod 29 = 6

- Define f :
**Z**_{m}x**Z**_{n}->**Z**_{mn}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.

- Determine the number of subgroups of
**Z**_{16}x**Z**_{17}. Justify your answer. - 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.