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\newcommand{\R}{\mathbb{R}} % real numbers
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\newcommand{\Q}{\mathbb{Q}} % rational numbers
% \newcommand{\st}{ \ : \ } % such that
\newcommand{\st}{ \ | \ } % such that
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\newcommand{\into}{\hookrightarrow}  % injection
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\title{MAT 150A HW06}
\author{[add your name here]}
\date{Due Tuesday, 2/20/24 at 11:59 pm on Gradescope}

\begin{document}

\maketitle

{
\scriptsize


\paragraph{Proof-based course}

This is a proof-based course and you are expected to \textbf{clearly prove} all your claims. If you're wondering how much detail to include, a good rule of thumb is that your proofs should be slightly more detailed than the proofs in the book, but not less detailed. They should also not be unreasonably verbose.


\paragraph{Reminder} 

Homeworks must be typed using LaTeX \textbf{in full sentences with proper mathematical formatting}. Handwritten homeworks will not be accepted. If there is a documented reason why you can't type up your homework, let me know and we can discuss an alternate policy. Otherwise, please consider learning how to properly write and typeset mathematics as part of this course.

}





%%%%%%%%%%%%%%%%%%%%%%%%
\exercise

Let $H$ and $K$ be subgroups of $G$.
Prove that if $HK = KH$, then $HK$ is a subgroup of $G$.


\solution




%%%%%%%%%%%%%%%%%%%%%%%%
\exercise

\bea
\item Let $C$ be a cyclic group generated by an element $g \in C$. Let $G$ be another group. Prove that any homomorphism $\varphi: C \to G$ is determined by the value of $\varphi$ on $g$.  
\item How many homomorphisms are there from $\Z \to \Z/15\Z$?
\item How many homomorphisms are there from $\Z/15\Z \to \Z$?
\ee

\solution




%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
% first isomorphism theorem
Use the First Isomorphism Theorem to find an isomorphism between $G/H$ and a more familiar group.
\bea
\item $G = \R^2$, $H = \R e_1$, where $e_1$ is the first standard basis vector. 
\item $G = \C^\times$, $H = \R^\times_{>0}$
\ee

\solution



%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
% first isomorphism theorem
Recall that the Klein four group is $V = \{1, a, b, ab\} = \langle a,b \st a^2 = b^2 = [a,b] = 1\rangle \cong C_2 \times C_2$ (see page 47 in the book).

\bea

\item Prove that the subgroup $N = \{ e, (1~2)(3~4), (1~3)(2~4), (1~4)(2~3)\}$ in $S_4$ is isomorphic to the Klein four group. 

\item Prove that $N$ is normal in $S_4$. \textit{Hint: Do not use brute force!}

\item Prove that the subgroup $H = \langle (1~2) , (3~4) \rangle  \leq S_4$ is also isomorphic to $V$, but is not a normal subgroup of $S_4$. 

\item Identify the quotient group $S_4/N$. That is, use the First Isomorphism Theorem to find an isomorphism from $S_4/N$ to a more familiar group. \textit{Hint: Use the counting formula first to figure out what the order of the quotient group is.}

\item How many subgroups are there in $S_4$ that contain $N$? \textit{Hint: Do not use brute force!}
\ee



\solution



%%%%%%%%%%%%%%%%%%%%%%%%
\exercise
Let $G$ be a group of order 21. Suppose it contains two \textbf{normal} subgroups $K$ and $N$, where $|K| = 3$ and $|N| = 7$. Prove that $G \cong K \times N$. 


\solution






\end{document}