Math 4281 – Homework 10
Textbook problems:
– Chapter 11: 9
– Chapter 16: 1abcefgh, 2, 9, 11, 17, 28, 31, 38
Note for Chapter 16, #1: In general R(x) means something different1 from R[x], and we
may or may not discuss the former in class, but in this problem it means the same thing as
R[x].
Other problems:
1. Suppose G is a group and K, H are two normal subgroups of G such that K ⊆ H.
(a) Prove that K is a normal subgroup of H.
(b) The set H/K = {hK : h ∈ H} is definitely a subset of G/K. Prove that it is actually
a normal subgroup of G/K.
(c) Prove that (G/K)/(H/K) ' G/H; this is the third isomorphism theorem. (Hint: Show
that the function φ : G/K → G/H defined by φ(gK) = gH is well-defined, surjective,
and has kernel H/K; then apply the first isomorphism theorem.)
1
What R(x) is is the set of rational functions in x, as opposed to R[x], the set of polynomials in x. Sometimes
these end up being the same thing, e.g. Q(i) = Q[i] because any expression (a + bi)/(c + di) can be rewritten
in the form a0 + b0 i.
1