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\title{Practice problems for Mid Term 2 \\ Abstract Algebra (5590H)}
\author{}
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\newcommand{\problem}[1]{\noindent {\bf Problem #1.}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\GL}{\mathrm{GL}}
\newcommand{\ds}{\displaystyle}
\newcommand{\Autg}{\mathrm{Aut}_{\scriptscriptstyle{\text{gp}}}}
\newcommand{\matthree}[9]{\left[\begin{array}{ccc}
#1 & #2 & #3\\
#4 & #5 & #6\\
#7 & #8 & #9
\end{array}\right]}
\begin{document}
\maketitle

\noindent {\bf Topics:} Sylow theorems \S 4.5. Direct and semidirect
products \S 5.1, 5.4 and 5.5. Finite abelian groups \S 5.2.
Automorphisms of groups \S 4.4. Alternating group \S 3.5, 4.6.\\

\problem{1} Let $G$ be a finite group with $231$ elements.
Prove that (a) there is a unique Sylow $11$--subgroup of
$G$ and it is contained in $Z(G)$ (b) there is a unique
Sylow $7$--subgroup.\\

\problem{2} List all finite abelian groups of size $300$.
In each case, give the invariant factors of the group.\\

\problem{3} Let $G$ be a finite group and $\psi : G\to G$
be an automorphism. Let $x\in G$ and $y=\psi(x)$. Show
that $\psi$ gives a bijection between conjugacy classes of
$G$ containing $x$ and $y$.\\

\problem{3}$\ast$ Take $G=S_5$ and let $\psi\in\Autg(S_5)$.
Use the previous exercise to prove the following. 
(i) $\psi$ gives a permutation of the set of $2$--cycles in $S_5$. 
(ii) There exists $a\in\{1,2,3,4,5\}$ such that
\[
\psi((1\ k)) = (a\ b_k),\ k=2,3,4,5.
\]
(iii) Let $\sigma\in S_5$ be given by $\sigma(1)=a$ and
$\sigma(k) = b_k$ ($k=2,3,4,5$). Then $\psi(x) = \sigma x \sigma^{-1}$.
(Hence, every automorphism of $S_5$ is inner).\\

\problem{4} Let $p\in\Z_{\geq 2}$ be a prime number,
and let $\mathbb{F}_p = \Z/p\Z$. Let $G$ be the following group
with $p^3$ elements:
\[
G = \left\{\matthree{1}{x}{z}{0}{1}{y}{0}{0}{1} : x,y,z\in\mathbb{F}_p\right\}.
\]
Show that $G\cong N\rtimes K$, where
\[
N = \left\{\matthree{1}{0}{z}{0}{1}{y}{0}{0}{1} : y,z\in\mathbb{F}_p\right\}
\qquad\text{and}\qquad
K = \left\{\matthree{1}{x}{0}{0}{1}{0}{0}{0}{1} : x\in\mathbb{F}_p\right\}.
\]

\problem{4}$\ast$ Note that $N\cong \mathbb{F}_p\times\mathbb{F}_p$
and $K\cong \mathbb{F}_p$. Compute the group homomorphism
$\alpha : \mathbb{F}_p \to \Autg(\mathbb{F}_p\times\mathbb{F}_p)$
so that $G\cong N\rtimes_{\alpha} K$.\\

\problem{5} Let $A_4 \triangleleft S_4$ be the alternating group.
Show that $A_4$ does not have a subgroup of size $6$.
(This is the standard counterexample to the converse of Lagrange's
theorem: $H<G$ implies $|H|$ divides $|G|$, but $k$ divides $|G|$ does
not imply that there is a subgroup of size $k$.)
\end{document}
