M417 Homework 8 Spring 2004
Instructions: Solutions are due Fri., April 9.
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.]
Find all solutions x to:
x mod 37 = 17
x mod 29 = 6
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.
Determine the number of subgroups of
Z16 x Z17. 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.