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

## Section2.2Preparation

###### ObjectivesLearning Goals for this Chapter
• State in words and in precise notation the properties of a group's binary operation.
• Use the definition to distinguish groups from non-groups.
• Identify the identity, simplify produts, and determine inverses of elements in given example groups.
• Construct the Cayley table for a given example group.
• Determine, with proof, whether a given example group is or is not abelian.

This chapter is our formal introduction to the algebraic structure known as a group, which is the main structure we study in first-semester abstract algebra. As we'll see, groups provide the minimal amount of structure that we need to "do algebra," i.e., simplify expressions and solve equations in ways that we will find familiar. In a sense, it is surprising that from so few axioms (associativity, closure, identity, inverse) we will discover that there is a lot to say about groups, some of which are still active areas of mathematical research today.

### Subsection2.2.1Acquaint 2

1. Watch the video above, and read Definitions and Properties 2.4 and leave comments, questions, and discussions using your http://hypothes.is account.
2. Complete Acquaint 2 by 10:00am on Monday 9/17.

### Subsection2.2.2Homework 2

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.

#### Subsubsection2.2.1Exercises

1. Exercises 2.5, #2.
2. Exercises 2.5, #6.
3. Exercises 2.5, #7.
4. Exercises 2.5, #10.
5. Exercises 2.5, #15.
6. Exercises 2.5, #16.
7. Exercises 2.5, #25.
8. Exercises 2.5, #26.
9. Exercises 2.5, #31.
10. In the group $U(9)\text{,}$ find the inverses of 2, 7, and 8.
11. Let $G$ be a group and $a,b,c \in G$ be elements. Solve each equation for $x\text{:}$ \begin{align*} axb \amp= c \amp a^{-1}xa \amp= c \end{align*}
12. Let $G$ be a group. Prove that every element of $G$ appears exactly once in each row of its Cayley table. Hint

There are two things to show:

1. Why does every $g \in G$ appear at least once in each row? (Try a direct proof.)
2. Why can't any $g$ appear more than once? (Try a proof by contradiction: what would happen if there existed a $g$ that appeared twice in a row?