Math 120A HW 9

These problems are assigned for practice only and are not to be turned in for grading.
  1. Let $G$ be a group, let $H$ be a normal subgroup of $G$, and let $a,b \in G$. List as many conditions as you can that are equivalent to $aH = bH$. Prove that your conditions are all equivalent.
  2. Let $\phi: G \to G'$ be a homomorphism of groups and define $H = \ker(\phi)$. Show that there is a well-defined function $\mu : G/H \to G'$ given by $\mu(aH) = \phi(a)$. (This is the first step in proving the fundamental homomorphism theorem.)
  3. Section 15 Exercises 1, 3, 5, 7, 9, 11. Hint: for Exercises 5 and 7 it helps to use a "nice" system of coset representatives. In both of these cases, you can show that tuples with $0$ in the first coordinate form a system of coset representatives, meaning that there is exactly one such tuple in every coset. Exercise 11 cannot (as far as I see) be solved by the techniques that we discussed in class, but it is still a reasonable problem for you to think about.
  4. Section 15 Exercise 19abcde.
  5. Section 15 Exercises 35, 36.