- Let H and N be subgroups of a group G, with N normal in G. Let I be the intersection of H and N. You may assume without proof the fact that I is normal in H and N is normal in HN (but of course you should be able to prove this fact if you need to!). The Second Isomorphism Theorem states that: H/I is isomorphic to HN/N. Prove the Second Isomorphism Theorem. [Hint: Define an appropriate homomorphism F: H -> HN/N and apply the First Isomorphism Theorem.]
- Let K and N be normal subgroups of a group G, with N contained in K. Use the FFF to show that K/N is a normal subgroup of G/N, and prove the Third Isomorphism Theorem (AKA the Freshman's Dream), that (G/N)/(K/N) is isomorphic to G/K. [Hint: Apply the FFF and the First Isomorphism Theorem.]
- Let g:
**Z**-> G and h:**Z**-> H be surjective homomorphisms of groups, where**Z**denotes the integers. Let |G| = m and |H| = n. Let F :**Z**-> GxH denote the homomorphism whose components are g and h.- (a) Find ker F and |Im(F)|. Justify your answers.
- (b) If gcd(m, n) = 1, show that
**Z**/mn**Z**is isomorphic to GxH. [Hint: apply (a), using the First isomorphism Theorem.] - (c) Given positive integers m and n such that gcd(m, n) = 1 and arbitrary integers a and b, prove the Chinese Remainder Theorem (CRT), that there is an integer c which is a simultaneous solution x = c to x = a mod m and x = b mod n (I'm using "=" since the usual triple equals is not really available in html), and that the set of all solutions is precisely the set of integers d such that mn | c - d. [Hint: Restate the CRT in terms of F.]

- Let m and n be relatively prime positive integers.
Given any integer i, let [i]
_{m}denote the coset i+m**Z**in**Z**(thus [i]_{m}is just the congruence class of i modulo m).- (a) You may assume that the map
g:
**Z**/mn**Z**->**Z**/m**Z**x**Z**/n**Z**defined by g([i]_{mn}) = ([i]_{m}, [i]_{n}) is an isomorphism. What is the inverse? (I.e., give a formula for where ([i]_{m}, [j]_{n}) must map to, given arbitrary integers i and j. You do not need to give proof, but be sure your answer really is the inverse.) - (b) You may assume that the map
h:
**Z**/m**Z**x**Z**/n**Z**->**Z**/mn**Z**defined by h(([i]_{m}, [j]_{n})) = [ni+mj]_{mn}is an isomorphism. What is the inverse? (I.e., give a formula for where [i]_{mn}must map to, given an arbitrary integer i. You do not need to give proof, but be sure your answer really is the inverse.)

- (a) You may assume that the map
g:
- This is #8, p. 75: if G is a group with normal subgroups of orders 3 and 5, prove that G has an element of order 15.
- (This is essentially #10.1, p. 76.)
Let G be the group of invertible real upper triangular 2x2 matrices.
For each of the following conditions, determine if it defines
a normal subgroup H of G (it is enough to just write "Yes" or "No"
without justification), and if so, determine G/H (it is enough
without justification to
define an explicit surjective homomorphism G -> K to some group K
such that the kernel is H; by the First Isomorphism Theorem, this
would show that G/H is isomorphic to K, just be sure that
your map really is an isomorphism with the required kernel!).

(a) a _{11}= 1(b) a _{12}= 0(c) a _{11}= a_{22}(d) a _{11}= a_{22}= 1