Math 360, Fall 2013, Assignment 11

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. Action (of a group $G$ on a set $X$).
2. $G$-set.
3. Homomorphism (of $G$-sets).
4. Isomorphism (of $G$-sets).
5. Orbit (in a $G$-set).
6. Transitive action.
7. Isotropy group (of a point in a $G$-set).
8. Ring.
9. Homomorphism (of rings).
10. Unity.
11. Unit (warning: this is not a synonym for "unity").
12. Division ring.
13. Field.
14. Subring.
15. Subfield.

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

1. Classification of transitive actions (we stated this in class; in the book it appears only as Exercise 16.15).

Solve the following problems:

1. Section 16, problems 2, 3, and 9.
2. Section 18, problems 3, 5, 7, 8, 11, 12, 14, and 17.