Math 360, Fall 2017, Assignment 13

"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "And then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision."

- Lewis Carroll, Alice's Adventures in Wonderland

Read:

1. Section 15.
2. Section 16.

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

1. Action (of a group $G$ on a set $S$).
2. $Gx$ (i.e. the orbit of the point $x$ under the action of the group $G$).
3. Transitive (action).
4. $G^x$ (i.e. the isotropy group of $x$).
5. Isomorphism (of group actions).

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

1. Fundamental theorem of homomorphisms.
2. Corollary relating the image, kernel, and domain of a homomorphism.

Solve the following problems:

1. Section 15, problems 1, 7, and 11 (in each problem, simply find a familiar group to which the given quotient group is isomorphic).
2. Section 16, problems 2 and 3.
3. (Action by conjugation) Let $G$ be any group. Show that $G$ acts on itself by the formula $g\cdot x = gxg^{-1}$. (This is called the action by conjugation, and its orbits are called the conjugacy classes of $G$.)
4. Describe the conjugacy classes of $S_3$ (see the previous problem for the definition).

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