$\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{\nmid} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\transpose}{\text{t}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \setcounter{chapter}{-1}$

## Section5.2Preparation

###### ObjectivesGoals
• Identify in context, and use "stack notation" and cycle notation to express, permutations of a finite set.
• Simplify products and determine inverses of permutations to illuminate the structure of $S_n\text{,}$ the group of permutations of $n$ symbols.
• Represent any permutation as a product of transpositions, and determine its sign (even/odd).
• Identify how properties of the alternating subgroup $A_n \lt S_n$ consisting only of even permutations compare to the properties of the full symmetric group $S_n\text{.}$
• Discuss the reason why any finite group is "the same" as a subgroup of some symmetric group.

### Subsection5.2.2Homework 5

Homework exercises are valuable preparation for your quiz on this material. Generally I do not collect homework for grading -- however, I will expect to review your homework progress in the event of your requesting a revision on a quiz. You are welcome to discuss homework problems with your groups and with the class, both in person and in the Slack #math channel.

#### Subsubsection5.2.1Exercises

1. Exercises 5.5, #1.
2. Exercises 5.5, #2.
3. Exercises 5.5, #3.
4. Exercises 5.5, #5. Hint

Since $S_4$ is not a cyclic group (why?), listing all its subgroups will take more effort and thinking than it did for cyclic groups. There are a lot of subgroups of order 2, for example...

5. Exercises 5.5, #8.
6. Exercises 5.5, #10. Hint

Begin by decomposing an arbitrary $\sigma$ into a product of disjoint cycles; this then boils down to the question "what's the largest possible least common multiple I can create?"

7. Exercises 5.5, #9.
8. Exercises 5.5, #11.
9. Exercises 5.5, #23. Hint

This is dangerously close to Proof Portfolio problem 2.3.3.

10. Exercises 5.5, #35. Hint

Note that the author means "for all" $\alpha,\beta\in S_n\text{,}$ the permutation $\alpha^{-1}\beta^{-1}\alpha\beta$ is even. The expression

\begin{equation*} \alpha^{-1}\beta^{-1}\alpha\beta \quad \text{is called the "commutator" of }\alpha,beta. \end{equation*}