Math 361, Spring 2014, Assignment 9

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

1. Solvable group.
2. $G$-set (this and related definitions are included for review, and can be found in Section 16 of the text).
3. Orbit (of a point in a $G$-set).
4. Isotropy group (at a point of a $G$-set).
5. Fixed point (of a $G$-set).

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

1. Theorem relating the cardinality of an orbit to the index of the isotropy group (Theorem 16.16 in the text).
2. Counting lemma for groups of prime power order (Theorem 36.1 in the text).
3. Cauchy's Theorem.
4. Sylow's First Theorem.
5. Theorem on solvability of groups of prime power order.

Solve the following problems:

1. Section 15, problem 39 (this is much too long to be the quiz question, but the fact that $A_n$ is a simple group for $n\geq5$ is a big deal, and this exercise will guide you through a proof).
2. Prove that any group of order 162 is solvable. (More generally, prove that for any prime $p$ and any positive integer $n$, any group of order $2p^n$ is solvable.)

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