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{\bf \centerline{Math 361 Exam 1;  Mon Oct 13, 10:40--11:30am}}\medskip
\noindent{\bf Instructions.} Answer questions 1--6.
 You must show all necessary working to receive full 
points, unless otherwise indicated.\medskip
\noindent{\bf 1.} {\sl (10 points)} Let $G=S_4$ denote the symmetric group 
$A(\{1,2,3,4\})$ on $4$ letters. Define the elements $g=\bigl({1\atop 2}{2\atop
3} {3\atop 4} {4\atop 1}\bigr)$ and $h=\bigl({1\atop 3} {2\atop 2} {3\atop 1}
{4\atop 4}\bigr)$ in $G$.\smallskip
 \item{(a)} Calculate $g^{-1}h$.\smallskip
\item{(b)} List the elements of the cyclic group generated by $g$. What is the
order of $g$?\medskip



\noindent{\bf 2.}  {\sl(20 points)}\smallskip  
\item{(a)} Give a careful definition of the term ``group.''\smallskip
\item{(b)} State whether the following are groups or not (no working is
required): (1) the set of real numbers with binary operation $*$ given by
$a*b=2(a+b)$ (2) the set of all positive real numbers under multiplication
(3) the set of all non-negative real numbers under addition.
 \medskip

\noindent{\bf 3.} {\sl (10 points)}
 Let $G$ be  a group.
\item{(a)} State formulae for  $(ab)^{-1}$ and $(a^{-1})^{-1}$,
for  $a,b\in G$. \smallskip 
 \item{(b)} For $x,y\in
G$, define the ``commutator'' $[x,y]$ of $x$ and $y$ by $[x,y]:=xyx^{-1}y^{-1}$.
 Show by direct calculation, using the formulae from (a), that
if $x,y,z\in G$, then
$$[zxz^{-1}, zyz^{-1}]=z[x,y]z^{-1}.$$\medskip

 

\noindent{\bf 4.} {\sl (20 points)} Let $G$ be the group of non-zero real
 numbers under
multiplication. (You do NOT have to prove that $G$ is a group).\smallskip 
\item{(a)} Prove that $H=\{\,2^n\mid n\in {\bold
Z}\,\}$ is a subgroup of $G$.\smallskip
\item{(b)} Let $n$ be a positive integer. Prove that the function $f\colon
G\rightarrow G$ given by $f(x)=x^n$ is a group homomorphism.\smallskip 
\item{(c)} Find the kernel and
image of $f$ (your answer to this part will depend on whether $n$ is even or
odd). For which values of $n$ is $f$ an isomorphism? 
 \medskip 

\noindent{\bf 5.} {\sl (20 points)}  (a) Explain what is meant by a right coset
$Hg$ of a subgroup $H$ of a group $G$. Define the index $[G\colon H]$ of $H$ in
$G$.\smallskip \item{(b)} State Lagrange's theorem, and explain briefly how
cosets are involved in its proof.\smallskip
\item{(c)} Is it possible for a group of order $75$ to have (1) a subgroup of
order $50$? (2) an element of order  $15$? (3) a subgroup of index 10? 
(Simply state yes or no for each part; no working is required).

\medskip \noindent{\bf 6.} {\sl (20 points)} Let $G={\bold Z}_{19}^*$
 be the group of units of the integers mod $19$, consisting
of congruence classes $[n]$ modulo $19$ with gcd$(n,19)$=1, under the
multiplication $[n][m]=[nm]$. (You do NOT have to prove $G$ is a group).
\smallskip

\item{(a)}  Determine whether $G$ is a cyclic group.
 \item{(b)} List the $6$ elements of the cyclic subgroup $H$ of $G$ generated by
$[8]$.
\item{(c)} Find all the distinct right cosets of $H$ in $G$. What is the index
$[G:H]$?
\item{(d)} Explain briefly why $H$ is a normal subgroup of $G$.
Write down the multiplication table for the quotient group $G/H$.  



\vfill\eject\bye




