\documentclass{amsart}
\title{Problems for practice}
\author{}

\usepackage[margin=1.25in]{geometry}
\usepackage{amsmath}
\usepackage{amssymb}

\newcommand{\problem}[1]{\noindent {\bf Problem #1.} }

\newcommand{\Autg}[1]{\operatorname{Aut}_{\scriptscriptstyle{gp}}\left(#1\right)}
\newcommand{\mult}[2]{\operatorname{Mult}(#1; #2)}

\newcommand{\Z}{\mathbb{Z}}

\begin{document}
\maketitle

\problem{1} Prove or disprove: $S_5$ is a semidirect product of $A_5$
and $\Z/2\Z$. {\em Recall: $S_n$ is the group of permutations of $\{1,2,\ldots,n\}$
and $A_n$ is the kernel of the sign homomorphism $\varepsilon : S_n \to \{\pm 1\}$.}\\

\problem{2} Determine whether the following statements are true or false.
\begin{enumerate}
\item If $G$ is a cyclic group, then $\Autg{G}$ is cyclic.\\

\item If $G$ is a cyclic group, then $\Autg{G}$ is abelian.\\

\item Let $N_1, N_2$ be two subgroups of a group $G$. Assume that $N_1$ is normal
in $G$ and $N_2\subset N_1$ is normal in $N_1$. Then $N_2$ is normal in $G$.\\

\item Every group $G$ admits at least one Jordan-H\"{o}lder series.
{\em Recall: a Jordan-H\"{o}lder series is a finite, strictly descending chain of
normal subgroups which cannot be further refined. Equivalently, each graded
piece must be simple.}\\

\end{enumerate}

\problem{3} Let $H$ and $N$ be two groups and $\alpha,\beta : H\to \Autg{N}$
two group homomorphisms. Consider the semidirect products $A=N\rtimes_{\alpha} H$,
and $B=N\rtimes_{\beta} H$.
\begin{enumerate}
\item Assume that there exists $T\in \Autg{N}$ such that $\beta(h)(n) = T(\alpha(h)(T^{-1}(n)))$
for every $h\in H$ and $n\in N$. Write an isomorphism
$F_T : A \to B$.\\

\item Assume that there exists $\psi\in \Autg{H}$ such that $\alpha(h) = \beta(\psi(h))$,
for every $h\in H$. Write an isomorphism $F_{\psi} : A\to B$.\\

\item Assume that there exists a group homomorphism $j : H\to N$ such that
\[\alpha(h)(n) = j(h)\cdot \left( \beta(h)(n)\right)\cdot \left(\beta(h)\left( j(h)^{-1}
\right)\right),
\]
for every $h\in H$
and $n\in N$. Write an isomorphism $F_j : A\to B$.\\
\end{enumerate}

\noindent {\bf Bonus.} Given any isomorphism $F : A\to B$, is it true that
we should be able to write $F$ (up to certain trivial identifications)
in terms of $F_T, F_\psi, F_j$ as above?\\

\problem{4} Let $G$ be a group and consider the group homomorphism 
$C : G\to \Autg{G}$ given by $C(g)(x) = gxg^{-1}$, for every $g,x\in G$.
Prove that $G\rtimes_C G$ is isomorphic to $G\times G$.\\

\problem{5} Let $H$ be any group with $27$ elements, and let $f:A_5 \to H$
be a group homomorphism. Prove that $f(x) = e$ (unit of $H$) for every $x\in A_5$.
{\em Recall: $A_5$ is simple.}\\

\problem{6} What does it mean for two composition series $\Sigma_1$
and $\Sigma_2$ of a group $G$ to be equivalent? (Just write the definition.)\\

\problem{7} Prove or disprove: two groups with equivalent Jordan-H\"{o}lder
series must be isomorphic.\\

\problem{8} What is the center of the group $G = \operatorname{GL}_2(\mathbb{F}_p)$?
Here $\mathbb{F}_p$ is the field with $p$ elements, where $p$ is a prime.\\

\problem{9} Compute the commutator series, and the central series
for the following group:
\[
B = \left\{\left(\begin{array}{ccc} d_1 & x & z \\ 0 & d_2 & y \\ 0 & 0 & d_3\end{array}\right)\ 
\text{ where } d_1,d_2,d_3\in \mathbb{C}\setminus\{0\},\ \text{and } 
x,y,z\in \mathbb{C} \right\}
\]

{\em Recall: commutator series is defined inductively as: $G^{(0)} = G$
and $G^{(\ell+1)} = [G^{(\ell)},G^{(\ell)}]$. Central series is defined inductively as
$C^1(G) = G$ and $C^{\ell+1}(G) = [G,C^{\ell}(G)]$.}\\

\problem{10} Write the Jordan-H\"{o}lder series for the dihedral group
$D_{12}$ with $12$ elements.\\

\problem{11} Assume that $G$ is a (non-trivial) nilpotent group. Prove that $Z(G)\not= \{e\}$.
Here $Z(G)$ is the center of $G$.
{\em Recall: $G$ is nilpotent iff $C^n(G) = \{e\}$ for some $n\geq 0$; iff
$G$ admits a composition series $G=H_0\triangleright \ldots \triangleright H_m=\{e\}$ such that
$[G,H_{\ell}]\subset H_{\ell+1}$ for every $\ell$.}\\

\problem{12} Prove that every $p$-group is nilpotent.
{\em Recall: a group $G$ is nilpotent iff $G/Z(G)$ is nilpotent.}\\

\problem{13} Give an example of a group $G$ and a normal subgroup $N\triangleleft G$
such that both $N$ and $G/N$ are nilpotent, but $G$ is not.\\

\problem{14} Compute $\Autg{\Z/2Z \times \Z/4\Z}$.\\

\problem{15} How many abelian groups of size $24$ are there?\\

\problem{16} Compute $\Autg{\Z/48\Z}$.\\

\problem{17} Let $G$ be a group and $N\triangleleft G$ be a normal subgroup.
Prove that $G/N$ is abelian if, and only if $[G,G]\subset N$.\\

\problem{18} True or false: center of a solvable group is always non-trivial.
{\em Recall: $G$ is solvable iff $G^{(n)} = \{e\}$ for some $n\geq 0$; iff
$G$ admits a composition series $G=H_0\triangleright \ldots \triangleright H_m=\{e\}$ such that
$H_{\ell}/H_{\ell+1}$ is abelian for every $\ell$.}\\

\problem{19} Let $B$ and $N\leq B$ be the following groups:
\begin{gather*}
B = \left\{ \left(\begin{array}{cc} d_1 & x \\ 0 & d_2\end{array}\right) \text{ where }
d_1,d_2\neq 0 \text{ and } x \text{ is arbitrary} \right\} \\
N = \left\{ \left(\begin{array}{cc} 1 & x \\ 0 & 1\end{array}\right) \text{ where }
x \text{ is arbitrary} \right\} 
\end{gather*}

Prove that $N$ is normal in $B$. Prove that $B$ is a semidirect product of $N$ and $H$,
where $H$ is the subgroup of $B$ consisting of diagonal matrices.\\

\problem{20} Prove or disprove: $\Autg{S_5} \cong S_5$.\\

\hrule\hrule

\vspace*{0.2in}

{\em Recall: every finite group admits a Jordan-H\"{o}lder series. Any two Jordan-H\"{o}lder
series are equivalent.}\\

\problem{21} For a finite group $H$, let $\ell(H)$ be the length of a Jordan-H\"{o}lder
series of $H$. How do we know that this number does not depend on the choice of 
the Jordan-H\"{o}lder series?\\

\problem{22} Let $G$ be a finite group and $N\triangleleft G$ be a normal subgroup.
Prove that $\ell(G) = \ell(N) + \ell(G/N)$.\\

\problem{23} Fix a finite simple group $S$. For a finite group $G$, choose a Jordan-H\"{o}lder
series $\Sigma: G = G_0\triangleright G_1 \triangleright \ldots \triangleright G_n = \{e\}$.
Let $\mult{S}{G}$ be defined as:
\[
\mult{S}{G} := \# \{ j : G_j/G_{j+1} \cong S\}
\]
Prove that the number $\mult{S}{G}$ does not depend on the choice of the Jordan-H\"{o}lder
series $\Sigma$.\\

\problem{24} Fix a finite simple group $S$. Let $G$ be a finite group and $N\triangleleft G$ be a normal subgroup.
Prove that $\mult{S}{G} = \mult{S}{N} + \mult{S}{G/N}$.\\

\problem{25} (Bonus). Prove or disprove: $\mult{S}{G}$ as defined above, can be computed
as cardinality of the set of group homomorphisms $S\to G$ counted up to conjugation by
elements of
$\Autg{S}$ and $\Autg{G}$. More precisely, is the following equation true?
\[
\mult{S}{G} = \left| 
\Autg{S} \backslash \operatorname{Hom}_{\scriptscriptstyle{gp}}(S,G)/ \Autg{G}
\right|
\]
\end{document}
