Assignment 4 – Due Friday, October 3
Turn this in at the start of recitation on Friday, October 3.
1. Read Section 2.7 of Herstein. (Judson covers similar material in Chapter 11.)
2. Herstein, p. 53: 9
3. Let A1 be the group of affine functions on R. (In Assignment 1, I made the mistake of defining
(f ∗ g)(x) = f (g(x)); from now on, we let the group operation on A1 be ◦, where (f ◦ g)(x) = g(f (x)).)
In the last assignment, you showed that H = {f1,b | b ∈ R} is a normal subgroup of A1 . Show that
A1 /H is isomorphic to (R∗ , ×), where (R∗ , ×) is the multiplicative group of nonzero real numbers.
4. Herstein, p. 64: 1, 2
5. (from Judson, p. 153, #14) Let G be a group and let G0 = haba−1 b−1 i; that is, G0 is the subgroup of
all finite products of elements in G of the form aba−1 b−1 . The subgroup G0 is called the commutator
subgroup of G.
(a) Show that G0 is a normal subgroup of G and that G is abelian if and only if G0 = {e}.
(b) Let N be a normal subgroup of G. Prove that G/N is abelian if and only if N contains the
commutator subgroup of G.
6. Let f : G1 → G2 be a homomorphism of groups. Show that:
(a) If H1 is a subgroup of G1 , then f (H1 ) is a subgroup of G2 . If H1 is normal, does f (H1 ) have to
be normal?
(b) If H2 is a subgroup of G2 , then f −1 (H2 ) is a subgroup of G1 . Furthermore, show that if H2 is
normal, then f −1 (H2 ) is normal.
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