Math 360, Fall 2019, 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

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

1. $aHa^{-1}$ (the conjugate of the subgroup $H$ by the element $a$).

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

1. Theorem giving several conditions equivalent to normality of a subgroup.

Solve the following problems:

1. Section 10, problems 20, 21, 22, 23, and 24.
2. Working in $S_3$, compute all conjugates of the subgroup $\left\langle(12)\right\rangle$. Do you notice any patterns in your answer?
3. Working in $S_3$, compute all conjugates of the subgroup $\left\langle(123)\right\rangle$.
4. Try to find a one-line argument that predicts the answer to the last problem without any calculations. (Hint: what is the index of the subgroup?)

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