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homomorphism, then $ker(\psi) \triangleleft G$.\\ @@ -263,7 +263,7 @@ \end{proof} -\begin{thm}{}(\emph{Second Isomorphism Theorem}) \label{2ndisothm} +\begin{thm}{i.2}(\emph{Second Isomorphism Theorem}) \label{2ndisothm} Let $H \subseteq G$, $N \triangleleft G$. Then \begin{enumerate}[i.] \item $HN \subseteq G$ @@ -329,7 +329,7 @@ \end{proof} -\begin{thm}{}(\emph{Third Isomorphism Theorem}) \label{2ndisothm}\\ +\begin{thm}{i.3}(\emph{Third Isomorphism Theorem}) \label{2ndisothm}\\ Let $H \subseteq K$ and $K \triangleleft G,~ H \triangleleft G$.\\ Then $\frac{K}{H} \triangleleft \frac{G}{H}$ @@ -388,6 +388,74 @@ \subsection{Chapter 14} +\begin{thm}{14.4} + $H \subseteq G,~~ N \triangleleft G$, then + \begin{enumerate} + \item if $G$ soluble $\Longrightarrow H$ soluble + \item if $G$ soluble $\Longrightarrow G/N$ soluble + \item if $N$ and $G/N$ soluble $\Longrightarrow G$ soluble + \end{enumerate} +\end{thm} +\begin{proof} + \begin{enumerate} + \item Since $G$ soluble, by definition: $\exists~~ 1=G_0 \triangleleft G_1 \triangleleft \ldots \triangleleft G_r = G$ with Abelian quotients $\frac{G_{i+1}}{G_i}$. + + Let $H_i = G_i \cap H$, then $H$ has a series $1=H_0 \triangleleft H_1 + \triangleleft \ldots \triangleleft H_r = H$, next we show that the + quotients $\frac{H_{i+1}}{H_i}$ are Abelian (so that H is soluble): + + $$\frac{H_{i+1}}{H_i} = \frac{G_{i+1} \cap H}{G_i \cap H} \stackrel{(*)}{=} \frac{G_{i+1} \cap H}{G_i \cap (G_{i+1}\cap H)} + \stackrel{(**)}{\cong} \frac{G_i(G_{i+1} \cap H)}{G_i} \subseteq \frac{G_{i+1}}{G_i} + $$ + + (*): to see why, $H_i = G_i \cap H = G_i \cap H_i = G_i \cap H_{i+1} = G_i \cap (G_{i+1} \cap H)$. + (**): by the 2nd Isomorphism Theorem (\ref{2ndisothm}). + + [TODO: diagram of subgroups] + + Notice that $\frac{G_{i+1}}{G_i}$ is Abelian, thus the left-hand-side of the congruence is also Abelian. + Therefore, $\frac{H_{i+1}}{H_i}$ is Abelian, thus $H$ is soluble. + + \item For $G/N$ to be soluble, (by definition) it would have the series $\frac{N}{N} = G_0 \frac{N}{N} \triangleleft G_1 \frac{N}{N} \triangleleft \ldots \triangleleft G_r \frac{N}{N} = \frac{G}{N}$, + and any quotient being $\frac{G_{i+1}\frac{N}{N}}{G_i \frac{N}{N}}$. + + The series clearly exists, so now we show that the quotients are Abelian, so that $G/N$ is soluble: + + $$ + \frac{G_{i+1} N}{G_i N} = \frac{G_{i+1}(G_i N)}{G_i N} \stackrel{*}{\cong} + \frac{G_{i+1}}{G_{i+1} \cap (G_i N)} \cong \frac{G_{i+1}/G_i}{(G_{i+1} + \cap (G_i N))/G_i} + $$ + + (*): by the 2nd Isomorphism Theorem (\ref{2ndisothm}). + + The last quotient is a quotient of the Abelian group $G_{i+1}/G_i$, so it is Abelian. + + Hence, $\frac{G_{i+1}N}{G_i N}$ is also Abelian; so $\frac{G}{N}$ is soluble. + + \item + By the definition of $N$ and $G/N$ being soluble, + \begin{align*} + N \text{soluble} \Longrightarrow~~ 1 &= N_0 \triangleleft N_1 \triangleleft \ldots \triangleleft N_r = N\\ + G/N \text{soluble} \Longrightarrow~~ 1= \frac{N}{N} &= \frac{G_0}{N} \triangleleft \frac{G_1}{N} \triangleleft \ldots \triangleleft \frac{G_r}{N} = \frac{G}{N} + \end{align*} + both with Abelian quotients. + + Consider the series of $G$ given by combining the two previous series: + + $$ + 1 &= N_0 \triangleleft N_1 \triangleleft \ldots \triangleleft N_r = N = G_0 \triangleleft G_1 \triangleleft \ldots \triangleleft G_r = G + $$ + the quotients are either + \begin{itemize} + \item $\frac{N_{i+1}}{N_i}$, Abelian + \item $\frac{G_{i+1}}{G_i}$, isomorphic to $\frac{G_{i+1}/N}{G_i/N}$, + which is Abelian. + \end{itemize} + \end{enumerate} + Therefore, the quotients are always Abelian; hence $G$ is soluble. +\end{proof} + \newpage \section{Tools}