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\begin{center}\begin{large}\textbf{Math 101 - Problem Set 6\\
Due Tuesday, October 24}\end{large}\end{center}



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\begin{enumerate} 

\item Determine whether $X$ along with the given binary operation $*$ is a group. If it is a group, show that all of the group axioms hold. If not, give a counterexample for one of the axioms.

\begin{enumerate}

\item $X$ is the set of functions from $\R$ to $\R$, and define $f*g = f\cdot g$ (multiplication).
\item $X$ is the set of functions from $\R$ to $\R$, and define $f*g = f\circ g$.
\item $X = \R_+$ and $a*b= \sqrt{ab}$.
\item $X = \{a/b \mid a \in \Z \textrm{ and } b \in \{1,2\}\}$ and define $x * y = x + y $.

\end{enumerate}

\item Find the order of each element of the group $\Z_{12}$.

\item Let $G$ be a group and suppose there is an element $a\in G$ of finite order $n$.

\begin{enumerate}

\item Show that $e, a^1, \ldots, a^{n-1}$ are all distinct elements of $G$. (Recall that $a^i = a * a * \cdots * a$ for $i$ factors $a$.)
\item If $G$ is finite, show that $|a| \leq |G|$.
\end{enumerate}

\item Let $G = \{e, a, b, c\}$. Come up with a binary operation $*$ on $G$ so that $G$ is a group with identity $e$ and no elements have order 4. Write out the $*$ table and show that you can't rename the elements to get $\lv \Z_4, + \rv$.

\item Let $G = \{a + b \sqrt{2} \in \R \mid a,b \in \mathbb{Q}\}$. 
\begin{enumerate}
\item Show that $G$ is a group under addition.
\item Show that $G - \{0\}$ is a group under multiplication.
\end{enumerate}

\item Let $\lv A, *_A\rv$ and $\lv B, *_B\rv$ be groups. Define the binary operation $*$ on $A \times B$ by $(a,b) * (c,d) = (a *_A c, b *_B d)$. Show that $\lv A\times B, * \rv$ is a group. The group $A \times B$ is called the \textit{\textbf{direct product}} of $A$ and $B$.

\item Show that $A \times B$ is abelian if and only if $A$ and $B$ are abelian.

\item Let $G$ be an abelian group and let $a,b \in G$ and $n \in \Zp$. Show that $(a * b)^n = a^n * b^n$. \\
\textit{(Hint: Induction!)}

\item Let $G$ be a group, and suppose $a^2 = e$ for all $a \in G$. Show that $G$ is abelian.

\end{enumerate}
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