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\noindent
\parbox{2.25in}{\bf Math 4720 \& 5210}
\hfill {\Large \bf Homework \#1} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Aug. 29${}^\mathbf{th}$\!, 2025}

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%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
%
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\begin{description}
\item[\#1 Establishing Order] Find the order and inverse of each of the following elements:

\begin{enumerate}[(a)]
\item Find the order and inverse of $(4,7)$ in $\mathbb{Z}_{5} \times \mathbb{Z}_{12}$. Also, what is the order of the group $\mathbb{Z}_{5} \times \mathbb{Z}_{12}$?

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{\it Note:} $\mathbb{Z}_n$ is a group under addition modulo $n$. You should be adding.

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\item Find the order and inverse of $(4,7)$ in $(\mathbb{Z}_{5})^\times \times (\mathbb{Z}_{12})^\times$. Also, what is the order of the group $(\mathbb{Z}_{5})^\times \times (\mathbb{Z}_{12})^\times$?

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{\it Note:} $(\mathbb{Z}_n)^\times = U(\mathbb{Z}_n) = \{ x \in \mathbb{Z}_n \;|\; \mathrm{gcd}(x,n)=1 \}$ (Gallian calls this $U(n)$) is the group of units of $\mathbb{Z}_n$. This is a group under multiplication modulo $n$. For example: $(\mathbb{Z}_{12})^\times = \{1,5,7,11\}$. You should be multiplying.

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\item Find the order and inverse of $\begin{bmatrix} 1 & 2 \\ 3 & 9 \end{bmatrix}$ in $(\mathbb{Z}_{10})^{2 \times 2}$. Also, what is the order of the group $(\mathbb{Z}_{10})^{2 \times 2}$?

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{\it Note:} $(\mathbb{Z}_m)^{n \times n}$ is a group under (matrix) addition with entries added modulo $m$. You should be adding.

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\item Find the order and inverse of $\begin{bmatrix} 1 & 2 \\ 3 & 9 \end{bmatrix}$ in $\mathrm{GL}_2(\mathbb{Z}_{10})$. 

{\it Note:} $\mathrm{GL}_2(\mathbb{Z}_{10}) = ((\mathbb{Z}_m)^{n \times n})^\times = \{ A \in (\mathbb{Z}_m)^{n \times n} \;|\; \mathrm{det}(A) \in (\mathbb{Z}_m)^\times \}$ is a group under (matrix) multiplication with entries computed modulo $m$. You should be multiplying.

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{\it Suggestion:} Even working mod 10, computing the order of this matrix can be tedious (doable but tedious). You might want to use some technology. For example, you could go to my SAGE sandbox\footnote{\href{https://billcookmath.com/sage/linear_algebra/}{https://billcookmath.com/sage/linear\underline{\  }algebra/}} and make SAGE do the heavy lifting. The following code creates a matrix $A$ in $(\mathbb{Z}_{10})^{2 \times 2}$. Then it prints both $A$ and $A^2$:

{\color{red}
\verb|A = matrix(IntegerModRing(10),[[1,2],[3,4]])|\\
\verb|pretty_print("A = ", A)|\\
\verb|pretty_print("A^2 = ", A^2)|
}

\item {\bf [Grad Problem:]} What is the order of the group $\mathrm{GL}_2(\mathbb{Z}_2)$? How about $\mathrm{GL}_2(\mathbb{Z}_{10})$? Then use the following boxed fact and these answers to find the order of $\mathrm{GL}_2(\mathbb{Z}_{10})$.

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\fbox{ \begin{minipage}{6.25in}
When $\mathbb{Z}_n$ is a field (i.e., when $n$ is prime), finding the order of $\mathrm{GL}_2(\mathbb{Z}_n)$ is relatively easy using some linear algebra. When $n$ is the product of distinct primes, we can then use the following:\footnote[14]{Juncheol Han's paper ``\href{https://koreascience.kr/article/JAKO200634741588239.page}{The general linear group over a ring}'', Bulletin of the Korean Mathematical Society, Vol. 43, No. 3 (2006).}

Suppose $k$ and $\ell$ are relatively prime. Then $\mathbb{Z}_{k\ell} \cong \mathbb{Z}_k \times \mathbb{Z}_\ell$ (as rings). Next, as rings we get $(\mathbb{Z}_{k\ell})^{2 \times 2} \cong (\mathbb{Z}_k)^{2 \times 2} \times (\mathbb{Z}_\ell)^{2 \times 2}$. It then follows, for their group of units, $\mathrm{GL}_2(\mathbb{Z}_{k\ell}) \cong \mathrm{GL}_2(\mathbb{Z}_{k}) \times \mathrm{GL}_2(\mathbb{Z}_{\ell})$. 
\end{minipage}}

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\end{enumerate}

%\item[\#2 Abel Approves] Use the classification of finitely generated Abelian groups.
%
%\begin{enumerate}[(a)]
%\item List all possible Abelian groups of order $4500 = 2^2\cdot 3^2 \cdot 5^3$ (up to isomorphism). Write each group using both invariant factors and elementary divisors.
%\item Let $G$ be an Abelian group of order $81=3^4$. (i) $G$ is not cyclic but does have an element of order $9$. What might $G$ be isomorphic to? (ii) What if $G$ does not have any elements of  order $9$?
%\end{enumerate} 

\item[\#2 Cyclical Issues] Draw a subgroup lattice for $\mathbb{Z}_{99}$. Then make a table listing possible element orders and how many elements there are of each of those orders. Finally, list all elements of order 9 in $\mathbb{Z}_{99}$.

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\item[\#3 Progressive Abstract Algebra -- Intersectionality] For each $i$ in some index set $I$, let $H_i$ be a subgroup of $G$. Prove that $\displaystyle \bigcap_{i \in I} H_i$ is a subgroup of $G$.

{\it Note:} Recall that $\displaystyle x \in \bigcap_{i \in I} H_i$ if and only if for every $i \in I$, we have $x \in H_i$.

\item[\#4 Lemprob Wardsback] Let $\varphi:G_1 \to G_2$ be a homomorphism between two groups. Let $K$ be a subgroup of $G_2$. Prove that $\varphi^{-1}(K)$ is a subgroup of $G_1$.

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{\it Note:} $\varphi^{-1}(K) = \{ x \in G_1 \;|\; \varphi(x) \in K \}$ is the {\it inverse image} (or {\it preimage}) of $K$. \hfill $x \in \varphi^{-1}(K)$ iff $\varphi(x) \in K$

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{\bf Warning:} Given some $x \in G_2$, ``$\varphi^{-1}(x)$'' is nonsense. The mapping $\varphi$ may not be invertible, so $\varphi^{-1}$ may not be defined {\it on elements}. On the other hand, $\varphi^{-1}(S)$ {\it always} makes sense for any subset, $S$, of the codomain. 

\item[\#5] \href{https://youtu.be/pZDUbKNiMsU}{\bf You're free cheesybread! You're free!} Use the universal property to show that each $x \in X$ is an element of infinite order in the free group $F(X)$. In particular, if $X$ is a non-empty set of generators, we have that $F(X)$ is an infinite group. 

{}{\bf [Grad Problem:]} Next, suppose $X = \{a,b,c\}$. Show that $ab^{-3}a^2c$ has infinite order in $F(X)$.

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 {\it Note:} Actually, {\it all} non-identity elements are of infinite order in $F(X)$. 

\end{description}

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