Math 360, Fall 2014, Assignment 10

The moving power of mathematical invention is not reasoning but the imagination.

- Augustus de Morgan

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

1. Simple group.
2. Group action.
3. Orbit (of a point in a $G$-set).
4. Isotropy (of a point in a $G$-set).
5. Homomorphism, monomorphism, epimorphism, isomorphism (of $G$-sets).
6. Transitive ($G$-set).

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

1. Fundamental theorem on homomorphisms.
2. Classification of transitive $G$-sets.
3. Lagrange-like theorem for orbits (Theorem 16.16).

Solve the following problems:

1. Section 15, problems 1, 11, and 36.
2. Section 16, problems 2 and 3.

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