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\begin{center} {\Large\bf Honors Algebra \\ Assignment 4} \\ Due,
Friday, Sept 25 \end{center}
\begin{quote}
I cannot live without people. -- Pope Francis
\end{quote}
\ben
\item In this problem, assume (important!) that $G$ is {\em Abelian}.
Set $H=\{g\in G: g^5=e\}$. (Warning: Expressions such as $x^{1/5}$ are
{\em not} well defined. Do not use them!)
\ben
\item Show $H$ is a subgroup of $G$. Point out where the assumption that
$G$ was Abelian was used.
\item Show $H$ is a normal subgroup of $G$. (Easy! -- If you see it.)
\item Assume further that $G$ is finite and that $H=\{e\}$. Show
that the map $\phi:G\ra G$ given by $\phi(g)=g^5$ is an automorphism.
(Definition: An automorphism is an isomorphism from a group to itself.)
\een
\item Let $G$ be any group and $g$ a fixed element of $G$. Define
$\phi: G\ra G$ by $\phi(x) = g^{-1}xg$.
\ben
\item Show that $\phi$ is a homomorphism.
\item Show that $\phi$ is an injection. To do this you have to
show that the equation $\phi(x)=e$ has {\em only} the solution
$x=e$.
\item Show that $\phi$ is a surjection. To do this you have to
show, given any $y\in G$, that the equation $\phi(x)=y$ has a
solution. [With these three you may conclude that $\phi$ is an
automorphism as defined above.]
\een
\item Find an isomorphism $\phi: (Z_{12},+)\ra (Z_{13}^*,\cdot)$.
(Idea: $\phi(1)$ determines $\phi$, try various $\phi(1)$ until
one works.)
\item Let $G$ be the positive reals under multiplication and let
$H$ be numbers $2^i$ where $i\in Z$.
\ben
\item Show $H$ is a subgroup of $G$.
\item Show $H$ is a Normal subgroup of $G$. (Easy! -- If you see it.)
\item Give a natural representation of the factor group $G/H$.
By this I mean that each element should be uniquely describable
as $\ol{a}$ where $a$ ranges over some natural set. (So, for
example, you couldn't have $3.1$ and $12.4$ as $\ol{3.1}=\ol{12.4}$.)
Have the identity represented as $\ol{1}$.
\item Find all elements $\ol{a}\in G/H$ whose cube (in $G/H$) is
the identity. (This is a bit tricky. There are precisely three
solutions!)
\een
\item Just for fun: What's purple and commutes?
\item Some questions about the order of an element.
\ben
\item Let $g\in G$ with $o(g)=100$. For what $i$ is $g^i=e$?
\item Let $g\in G$ with $g^{100}=e$. What are the possible
values of $o(g)$?
\item Let $\phi:G\ra H$ be a homomorphism and let $g\in G$
and set $h=\phi(g)$. Suppose $o(h)=100$. Assume $g$ has
finite order. What are the
possible values of $o(g)$?
\een
\een
\begin{quote}
The world can be divided into those who love New York City and those
who don't. Those who love New York tend to be unusually lively people.
They have to be. Characteristically, they are ambitious, curious,
intellectually vigorous, culturally alive. Such people give New
York City institutions great dynamism and some eccentricity.
\\ James Hester 1924-2015, NYU President
\end{quote}
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