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

## Section2.3Proof Portfolio Problems

###### 1

Let $S = \bigl\{ x \in \mathbb{R} \colon x \gt 6 \bigr\}$ and define a binary operation on $S$ by

\begin{equation*} a \star b = (a-1)(b-5) + 1 . \end{equation*}

Prove that $(S,\star)$ satisfies the closure, identity, and inverse properties. Furthermore, determine (with proof) whether $(S,\star)$ is a group.

###### 2

Let $G$ be a group and $g\in G$ be an element. Prove that the order of $g$ agrees with the order of its inverse:

\begin{equation*} \bigl| g \bigr| = \bigl| g^{-1} \bigr|. \end{equation*}
###### 3

Let $G$ be a finite cyclic group whose order is even. Let $g \in G$ be an element whose order is odd.

Prove that the order of $g$ agrees with the order of its square:

\begin{equation*} \bigl| g^2 \bigr| = |g|. \end{equation*}
###### 4

Let $G$ be a finite group whose order is even. Define a relation $\sim$ on $G$ by

\begin{equation*} a \sim b \text{ if } \begin{cases}a=b \amp {\rm or}\\ ab=e\end{cases}. \end{equation*}

Prove that $\sim$ is an equivalence relation. Then, use this equivalence relation to prove that there must exist an element of order 2 in the group $G\text{.}$