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\textbf{Name:} \smallskip  \\
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\begin{center}
{
Math 2106, Foundations of Mathematical Proof \\
HW 7 --- Due April 21 (Friday)
}
\end{center}

\noindent Reference: {\em Abstract Algebra} by Thomas Judson. 
\url{http://abstract.pugetsound.edu}

\begin{enumerate}
\item Prove that composition of functions is associative.  That is, for any functions \mbox{$f: A\rightarrow B$}, $g: B \rightarrow C$, and $h:C\rightarrow D$ show that $h \circ (g \circ f) = (h \circ g) \circ f$.
\item Let $S = \RR - \{-1\}$ and define a binary operation on $S$ by $a * b = a+b+ab$.  Prove that $(S,*)$ is an abelian group.
\item Prove that the set 
\[
S = \left\{\begin{pmatrix}1&x&y \\ 0&1&z\\0&0&1\end{pmatrix} : x,y,z \in \RR 
\right\}
\]
form a non-abelian group under matrix multiplication, called the {\em Heisenberg group}.
\item Write out the Cayley table for the group $(\ZZ_4,+)$.
\item Let $U(8) = \{[1], [3], [5], [7]\}$ where $[i]$ denotes the equivalence class of $i$ modulo $8$.  Write out the multiplication table for $U(8)$ under the operation $[a] \cdot [b] \mapsto [ab]$. Prove that $U(8)$ is a group under this operation.  Is $U(8)$ the same as $\ZZ_4$ up to relabeling? \\ (The group $U(8)$ is called $\ZZ_8^*$ in some textbooks.)
\item Write out the Cayley table for the group $G$ formed by the symmetries of a rectangle that is not a square. Is $G$ the same as either $\ZZ_4$ or $U(8)$ up to relabeling?
\item Show that  up to relabeling there are exactly two different groups of order $4$.  \\(One is called the {\em cyclic group of order $4$} and the other is called the {\em Klein four-group}.)
\item Let $G$ be a finite group with identity $e \in G$.  
\begin{enumerate}
\item Prove that for each $g \in G$, there exists an integer $n > 0$ such that $g^n = e$.
\item Prove that there exists an integer $m > 0$ such that $g^m = e$ for all $g \in G$.
\end{enumerate}
Note: $g^k = \underbrace{g*\cdots*g}_{k\text{ times}}$.
\item Prove that the inverse of $g_1 g_2 \cdots g_n$ in a group is $g_n^{-1}\cdots g_2^{-1} g_1^{-1}$.
\item Prove that if $g^2 = e$ for all $g \in G$, then $G$ is abelian.
\item Prove that a group $G$ is abelian if and only if $(gh)^2 = g^2h^2$ for all $g,h \in G$.
\item  Prove that a group $G$ is abelian if and only if $(gh)^{-1} = g^{-1} h^{-1}$ for all $g,h \in G$.
\item
\begin{enumerate}
\item Give an example of an infinite group $G$ and an infinite subgroup $H \subsetneq G$ such that $[G:H]$ is finite.
\item Give an example of an infinite group $G$ and an infinite subgroup $H \subsetneq G$ such that $[G:H]$ is infinite.
\end{enumerate}
\item Let $G$ be a group and $H$ be a subgroup.  Suppose $ghg^{-1} \in H$ for all $g \in G$ and $h \in H$.  Show that left cosets and right cosets coincide, that is, $gH = Hg$ for all $g \in G$.
\end{enumerate}


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