\magnification=\magstep1
\centerline{Mathematics 243, section 1 -- Algebraic Structures}
\centerline{Information on Exam 3}
\centerline{November 24, 2003}
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\noindent
{\it General Information}
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The third exam this semester will be
given the Friday after Thanksgiving, December 5.  The exam will cover the
material we have discussed since the second exam, through
Cayley's Theorem from class on Monday, November 24 -- the material
from Chapters 3, 4 (Groups).
in Gilbert and Gilbert.  The topics to review are:
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\item{1)}  The definition of a group; how to determine if a given 
set with a given binary operation has the structure of a group; key 
examples such as ${\bf Z}_m, +$, the group of invertible
classes in ${\bf Z}_m$ under multiplication:  ${\bf Z}_m^\times, \cdot$, 
the group of invertible $2\times 2$ matrices, $GL(2,{\bf R})$
under the matrix product, groups of permutations, 
etc., what it means for a group to be abelian,
which of the key examples are abelian and which are not.  
\item{2)}  Subgroups, the general subgroup criterion from Theorem 3.9,
cyclic subgroups.
\item{3)}  Cyclic groups, generators, orders of elements, the 
distinct subgroups of a cyclic group of order $n$.
\item{4)}  Isomorphisms of groups.
\item{5)}  Group homomorphisms, the kernel, other properties.
\item{6)}  Permutation groups, cycle decompositions, even and 
odd permutations, Cayley's Theorem.
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\noindent
{\it Proofs to Know}
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\item{1)}  Know how to prove Theorem 3.9 and how to apply it to 
show that subsets of groups are or are not subgroups, both in 
explicit examples and in cases where the subset is defined by 
a condition such as we saw in the problems about the center of a group,
the conjugate subgroups $aHa^{-1}$, the kernel of a homomorphism, etc.
\item{2)}  Let $G = \langle a\rangle$ be a cyclic group.  Then
\itemitem{a)}  Every subgroup $H$ of $G$ is cyclic.
\itemitem{b)}  If $G$ is finite cyclic of order $n$ and 
$H = \langle a^k\rangle$, then $H = \langle a^d \rangle$ where $d = \gcd(n,k)$.
\itemitem{c)}  If $G$ is finite cyclic of order $n$, then $a^k$ is 
a generator of $G$ if and only if $\gcd(k,n) = 1$.
\item{3)}  If $\varphi : G \to H$ is a group homomorphism, then
$\varphi(e_G) = e_H$, and for all $x\in G$, 
$\varphi(x^{-1}) = (\varphi(x))^{-1}$.
\item{4)}  The kernel of a group homomorphism is a subgroup 
of the domain.
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\noindent
{\it Some Review Problems}
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From Gilbert and Gilbert:
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\item{1)} Section 3.1: problems like 1-33, 44, 48, 49.
\item{2)} Section 3.2: 9, 10, 12, 13, 16, 23, 24, 35 (``closed''
means closed under the group operation from $G$) 
\item{3)} Section 3.3: 7, 8, 9, 11, 19, 20, 26, 27, 29, 30, 33
\item{4)} Section 3.4: 11, 16,17, 18
\item{5)} Section 3.5: 5,7,11,12 
\item{6)} Section 4.1: problems like 1-9, 15
\item{7)} Section 4.2: 5
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\noindent
{\it Review Session}
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I have off-campus commitments in Boston on the evenings of Wednesday,
December 3, and Thursday, December 4.  I will need to leave campus
at 5:30 pm at the latest each of those days. Thus, if we are going to do 
an evening review session this time, it has to be Monday, December  1
or Tuesday, December 2.
I am happy to do a session either evening, but be aware that the better 
prepared you are that evening, the more valuable the session will be.
{\it A word to the wise: Do not put off studying for this exam until 
after the review session!}
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\noindent
{\it Sample Exam}
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\noindent
I.  Let ${\bf Q}$ be the set of rational numbers:
${\bf Q} = \{m/n : m,n \in {\bf Z}, n \ne 0\}$.
Define a binary operation $*$ on ${\bf Q}$ by
$x * y = x + y + x\cdot y$
(where $\cdot$ is ordinary multiplication).  Is ${\bf Q}$ a group under $*$?
Why or why not?
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\noindent
II.  Find all generators the group $G = {\bf Z}_{21}$,
in which the operation is addition mod 21.  
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\noindent
III.  Is the symmetric group $S_3$ cyclic?  Why or why not?
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\noindent
IV.  Let $G = \langle a\rangle$ be a cyclic group.
\item{A)}  Show that every subgroup $H \subset G$ is cyclic.
\item{B)}  Show that if $G$ is finite, with $|G| = n$, then
$\langle a^k\rangle = \langle a^d\rangle$ where $d = \gcd(n,k)$.
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\noindent
V.  Let $G = {\bf Z}_{12}$ and $H = {\bf Z}_9$, which are both
groups under addition.  We write $[x]_{12}$ for the congruence class
of $x$ mod 12, and similarly $[x]_9$ for the class mod 9.  Define 
$\phi : G \to H$ by $\phi([x]_{12}) = [3x]_9$.
\item{A)} Show that $[x]_{12} = [y]_{12}$ implies
$[3x]_9 = [3y]_9$ (so that this mapping actually makes sense).
\item{B)} Show that $\phi$ is a {\it group homomorphism\/}.
\item{C)} Find all the elements in $\ker(\phi)$.
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\noindent
VI.  Let $G$ be a group and let $a \in G$ be a fixed element.  Define
$$C(a) = \{x \in G : ax = xa\}$$
\item{A)} Is $b = \pmatrix{1&2\cr 3&4\cr}$ in $C(a)$ for
$a = \pmatrix{1&1\cr 0&1\cr}$ in $G = GL(2,{\bf R})$ (a group under
matrix multiplication)?  Why or why not?
\item{B)} Show that $C(a)$ is a subgroup 
of $G$.   
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\noindent
{\it Extra Credit}  (10) Can a group $G$ with 5 elements contain an element
$a$ of order 3?  Why or why not?
\bye
