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\textbf{College of the Holy Cross, Fall Semester, 2018} \\
\textbf{MATH 351, Modern Algebra I -- Review for Final Exam} \\
\textbf{November 28}
\end{center}

\noindent
{\it General Information, Format and Groundrules}
\vskip 10pt
The final exam for this course will be given during the 
regular scheduled period for 8:00am MWF classes: 8:00am to 10:30am
on Friday, December 14.   
This will be an {\it closed book} exam.  
The exam will be written so that if you are well prepared and 
work steadily, you should take about 100 minutes hours to complete it, 
but you will have the full exam period (150 minutes) to work on it.

{\it As I indicated to the class after the first midterm, I will compute
a subscore (out of 100 possible points) on questions covering the material from the 
first midterm.  If that subscore is higher than your
grade on the first midterm, I will use it to replace the
first midterm grade in computing your overall course
average.}

\vskip 10pt
\noindent
{\it Topics}
\vskip 10pt
The final exam will be a {\it comprehensive exam}.  It will
cover all of the material we have discussed since the beginning of
the semester, giving roughly equal attention (roughly 2/5 of the possible points)
to each of the sections of the class corresponding to the two midterms and roughly 1/5 of
the possible points on the material on the symmetric and alternating groups, 
the class equation, and the Sylow Theorems after the second midterm.  (Of course,
many things from the second midterm and the new material also depend on earlier material!)

Like the midterm exams, this one will
focus mainly on the key definitions and theorems we have studied.
There will also be a few problems or parts of problems
where you will need to apply some of those definitions
and theorems to new situations.

\vskip 10pt
\noindent
Some more specific information:
\begin{enumerate}
\item All ``prove Theorem X''-type questions will be ones from 
the lists of proofs to know from the review sheets for the two midterms.
\item The new topics to be covered, {\it in addition to what was included
on the two midterms}, are:
\vskip 10pt
\begin{enumerate}
\item  The disjoint cycle decomposition of permutations in $S_n$; the
decomposition of permutations into transpositions ($2$-cycles), the subgroup
$A_n \subset S_n$.  Know which permutations are in $A_n$, how to compute
orders of permutations, and what the conjugacy classes in $S_n$ are.
 {\it According to the first point above, I will not ask you to show that $A_5$ is a simple
group}.
\item  The Sylow theorems and the background needed for their proof -- 
the conjugacy relation on a group, the class equation of $G$ and the role of the 
center $Z(G)$, centralizers of $a \in G$, the formula 
$|C_a| = [G:C(a)] = |G|/|C(a)|$ (that is: the order of the conjugacy class
of $a$ is equal to the index of the centralizer of $a$), 
normalizers of subgroups, and so forth.  {\it As noted in the first point above,
I will not ask you for the proofs, 
but you should know the statements of the three Sylow theorems}.  
\item  Applications of the Sylow theorems to structure of finite groups.
\end{enumerate}
\end{enumerate}
\vskip 10pt
\noindent
Here's a good ``computational'' review problem that covers a lot of what we did this semester.
The parts covering material from the first midterm are marked with (*).
(In fact, I'm considering putting something like this on the final itself!)
Consider the following operation table:
\[
\begin{array}{cccccccccccccc}
\cdot & \vline & g_1 & g_2 & g_3 & g_4 & g_5 & g_6 & g_7 & g_8 & g_9 & g_{10} & g_{11} & g_{12} \\ \hline
g_1 & \vline & g_1 & g_2 & g_3 & g_4 & g_5 & g_6 & g_7 & g_8 & g_9 & g_{10} & g_{11} & g_{12} \\
g_2 & \vline & g_2 & g_1 & g_4 & g_3 & g_6 & g_5 & g_8 & g_7 & g_{10} & g_9 & g_{12} & g_{11} \\
g_3 & \vline & g_3 & g_4 & g_7 & g_8 & g_{11} & g_{12} & g_1 & g_2 & g_5 & g_6 & g_9 & g_{10} \\
g_4 & \vline & g_4 & g_3 & g_8 & g_7 & g_{12} & g_{11} & g_2 & g_1 & g_6 & g_5 & g_{10} & g_9 \\
g_5 & \vline & g_5 & g_6 & g_9 & g_{10} & g_1 & g_2 & g_{11} & g_{12} & g_3 & g_4 & g_7 & g_8 \\
g_6 & \vline & g_6 & g_5 & g_{10} & g_9 & g_2 & g_1 & g_{12} & g_{11} & g_4 & g_3 & g_8 & g_7 \\
g_7 & \vline & g_7 & g_8 & g_1 & g_2 & g_9 & g_{10} & g_3 & g_4 & g_{11} & g_{12} & g_5 & g_6 \\
g_8 & \vline & g_8 & g_7 & g_2 & g_1 & g_{10} & g_9 & g_4 & g_3 & g_{12} & g_{11} & g_6 & g_5 \\
g_9 & \vline & g_9 & g_{10} & g_{11} & g_{12} & g_7 & g_8 & g_5 & g_6 & g_1 & g_2 & g_3 & g_4 \\
g_{10} & \vline & g_{10} & g_9 & g_{12} & g_{11} & g_8 & g_7 & g_6 & g_5 & g_2 & g_1 & g_4 & g_3 \\
g_{11} & \vline & g_{11} & g_{12} & g_5 & g_6 & g_3 & g_4 & g_9 & g_{10} & g_7 & g_8 & g_1 & g_2 \\
g_{12} & \vline & g_{12} & g_{11} & g_6 & g_5 & g_4 & g_3 & g_{10} & g_9 & g_8 & g_7 & g_2 & g_1 \\
\end{array}
\]
\noindent
You may assume without proof that this is the operation table for a group $G$ of order 12. 
\begin{enumerate}
\item[a.] (*) Which is the identity element in this group?  Which element is the inverse of each element? Is
$G$ abelian?  
\item[b.] (*) What are the orders of each element in this group?
\item[c.] (*) Why is $H = \{g_1,g_2,g_{11},g_{12}\}$ a subgroup of $G$?  
\item[d.] To which ``standard'' group of order $4$ is $H$ isomorphic?
\item[e.] (*) Why is $J = \{g_1,g_3,g_5,g_7,g_9,g_{11}\}$ a subgroup of $G$?  
\item[f.] To which ``standard'' group of order 6 is $J$ isomorphic?
\item[g.] (*) Why is $K = \{g_1,g_3,g_7\}$ a subgroup of $G$? 
\item[h.] (*) Which of the subgroups $H,J,K$ from the previous parts is normal in $G$?
\item[i.] For each subgroup that is normal in $G$ from the previous part, construct the factor group.
(That is construct $G/H$ if $H$ is normal, $G/J$ if $J$ is normal in $G$, and $G/K$ if $K$ is normal).  
\item[j.] Construct a group homomorphism $\alpha : G \to L$ for some group $L$ to make 
$\ker(\alpha) = \{g_1,g_2\}$.  To which ``standard'' group of order $6$ is $L$ isomorphic?  (Hint: To
see how to define $\alpha$, you might note the way 
the whole table for $G$ breaks up into $2\times 2$ blocks according to the cosets of the subgroup
$\{g_1,g_2\}$!)
\item[k.] Show that $G$ is the internal direct product of its subgroups $N = \{g_1,g_2\}$
and $J = \{g_1,g_3,g_5,g_7,g_9,g_{11}\}$.
\item[l.] What is $Z(G)$, the center of $G$?  What is the centralizer of the element $g_2$?  
\item[m.] What are the conjugacy classes in $G$?
\item[n.] How many different Sylow $2$-subgroups are there in $G$?  How many different Sylow $3$-subgroups?
Verify that the statement of Sylow III holds here.  
\end{enumerate}
\vskip 10pt
\noindent
For more review problems, also consult the review sheets for the two midterms and:
\begin{itemize}
\item 6.21, 6.23, 6.25
\item 7.1, 7.7, 7.9, 7.11, 7.17, 7.21, 7.29, 7.31, 7.37
\end{itemize}
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