Math 361, Spring 2021, Assignment 1
From cartan.math.umb.edu
Read:
- Section 13.
Carefully define the following terms, then give one example and one non-example of each:
- Homomorphism.
- Monomorphism.
- Epimorphism.
- Pushforward (of a subgroup under a homomorphism; also known as the forward image of the subgroup).
- Pullback (of a subgroup under a homomorphism; also known as the pre-image of the subgroup).
- Image (of a homomorphism).
- Kernel (of a homomorphism).
- Canonical projection (from a group G to a quotient G/H).
Carefully state the following theorems (you do not need to prove them):
- Theorem characterizing pushforwards ("The pushforward of a subgroup is a...").
- Theorem concerning pullbacks ("The pullback of a subgroup is a...").
- Theorem relating images to surjectivity.
Solve the following problems:
- Section 13, problems 1, 3, 5, 9, 28, 29, 45, and 49.
- Prove the theorem concerning pullbacks.
- Suppose that G and K are groups and ϕ:G→K is a homomorphism. Prove that for any H≤G, we have H⊆ϕ−1[ϕ[H]].
- Give an example to show that strict containment may occur in the result above. (Hint: start with a homomorphism which is not injective.)
- Suppose that G and K are groups and ϕ:G→K is a homomorphism. Prove that for any L≤K, we have ϕ[ϕ−1[L]]⊆L.
- Give an example to show that strict containment may occur in the result above. (Hint: start with a homomorphism which is not surjective.)