Assignment 6 – Due Friday, October 17
Turn this in at the start of recitation on Friday, October 17.
1. Read Section 2.9 of Herstein.
2. Suppose that H and N are normal subgroups of G, that H ∩ N = {e}, and HN = G.
(a) Show that for every g ∈ G, there are unique elements h ∈ H, n ∈ N such that g = hn.
(b) Show that hn = nh for every h ∈ H and n ∈ N . (Hint: Show that hnh−1 n−1 ∈ H and
hnh−1 n−1 ∈ N .) By the theorem in class, this implies that G ∼
= H × N.
3. Show that Z12 is isomorphic to Z3 × Z4 . Is Z9 isomorphic to Z3 × Z3 ? Why or why not?
4. List all the homomorphisms from Z/6Z = Z6 to Z/9Z = Z9 . How are these homomorphisms related
to multiplication by 0, 3, and 6? Why isn’t f (6Z + n) = 9Z + 2n a homomorphism from Z/6 to Z/6,
even though g(n) = 2n is a homomorphism from Z to Z?
5. In class, I asked the question:
Suppose f : G1 → G2 is a homomorphism, N1 / G1 , and N2 / G2 . When does f induce a
homomorphism f¯ : G1 /N1 → G2 /N2 ?
In this problem, we will answer that question.
(a) Suppose G1 , G2 are groups and that N1 ⊂ G1 , N2 ⊂ G2 are normal subgroups. Let f be a
homomorphism. When is f¯(N1 g) = N2 f (g) a well-defined homomorphism G1 /N1 to G2 /N2 ?
When f¯ is a well-defined homomorphism, we call it the homomorphism induced by f .
(b) Let G1 = G2 = R and let N1 = N2 = Z. For every t ∈ R, the map ft : R → R given by ft (x) = tx
is a homomorphism. When does ft induce a homomorphism f¯t : R/Z → R/Z? Give a formula for
f¯t . Graph f¯2 on the axes below:
Z + .8
Z + .6
Z + .4
Z + .2
Z
Z
Z + .2
Z + .4
Z + .6
Z + .8
(c) Show that if f is surjective, then f¯ is surjective (onto). Show that if f −1 (N2 ) = N1 , then f¯ is
injective (one-to-one).
(d) Suppose that K and N are normal subgroups of G and K ⊂ N . Use part (c) to show that the
canonical homomorphism π : G → G/K induces an isomorphism π̄ : G/N → (G/K)/(N/K).
Give an example of three groups G, N , and K that satisfy the hypotheses and describe the
isomorphism π̄.
6. Herstein: p. 70, #6, 10
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