Math 361, Spring 2013, Assignment 8

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

1. Subnormal series.
2. Isomorphism (of two subnormal series---see Definition 35.6).
3. Refinement (of a subnormal series).
4. Composition series.
5. Composition factors.
6. Solvable group.
7. \(G\)-set.
8. Orbit.
9. Isotropy subgroup.

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

1. Schreier Refinement Theorem (Theorem 35.11 in the text).
2. Jordan-Hölder Theorem (Theorem 35.15 in the text).
3. Classification of transitive \(G\)-sets (conclusion of exercise 16.15).
4. Theorem on the cardinality of \(G\)-sets when the order of \(G\) is a power of a prime (Theorem 36.1 in the text).

Do the following problems:

1. Section 35, problems 1, 3, 9, and 19.
2. Section 16, problems 1, 2, 3, 9, and 15.