Math 360, Fall 2016, Assignment 12

Algebra begins with the unknown and ends with the unknowable.

- Anonymous

Carefully define the following terms, then give one example and one non-example of each:

1. Pushforward (or forward image, or just image) of a set under a function.
2. Pullback (or pre-image or inverse image) of a set under a function.
3. Image (of a group homomorphism).
4. Kernel (of a group homomorphism).

Carefully state the following theorems (you do not need to prove them):

1. Theorem concerning pushforwards and pullbacks of subgroups.
2. Test for surjectivity of a homomorphism (in terms of its image).
3. Test for injectivity of a homomorphism (in terms of its kernel).

Solve the following problems:

1. Section 13, problems 16, 17, 19, 20, and 47.

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