\documentclass[12pt]{amsart}
\usepackage{amsmath,amsthm,amsfonts,latexsym,amssymb}
\pagestyle{headings}
\setlength{\textwidth}{36true pc}
\setlength{\headheight}{8true pt} % needed for 12pt documents
\setlength{\oddsidemargin}{15 truept}
\setlength{\evensidemargin}{15 truept}
\setlength{\textheight}{572true pt}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lm}[thm]{Lemma}
\newtheorem{prp}[thm]{Proposition}
\newtheorem{clm}[thm]{Claim}
%
\theoremstyle{definition}
\newtheorem{df}[thm]{Definition}
\newtheorem{exmp}[thm]{Example}
\newtheorem{prb}[thm]{Problem}
%
\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark}
%
\numberwithin{equation}{section}
%
\newcommand{\thref}[1]{Theorem~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lmref}[1]{Lemma~\ref{#1}}
%
\DeclareMathOperator{\satf}{\models}
\DeclareMathOperator{\miff}{\Longleftrightarrow} 
                                           % math `if and only if'
\DeclareMathOperator{\mimplies}{\Longrightarrow} 
                                           % math `implies'
\DeclareMathOperator{\mimplied}{\Longleftarrow}  
                                           % math `implied by'
\newcommand{\set}[2]{\{#1\,:\,\text{#2}\}} % { x : ... }
\DeclareMathOperator{\id}{\textsl{id}}     % The identity map
\DeclareMathOperator{\miffdef}{\stackrel{def}{\miff}}
                                           % defining iff
\DeclareMathOperator{\eqdef}{\stackrel{def}{=}}
                                           % defining equation

\DeclareMathOperator{\neareq}{\stackrel{\ae}{=}}
\DeclareMathOperator{\aeqv}{\stackrel{ae}{\equiv}}
\DeclareMathOperator{\aeq}{\stackrel{ae}{=}}
%
\DeclareMathOperator{\BZ}{\mathbb{Z}} % integers
\DeclareMathOperator{\BQ}{\mathbb{Q}} % rationals
\DeclareMathOperator{\BR}{\mathbb{R}} % reals
\DeclareMathOperator{\BC}{\mathbb{C}} % complex numbers
\DeclareMathOperator{\BE}{\mathbb{E}} % 
\DeclareMathOperator{\BF}{\mathbb{F}} % 
\DeclareMathOperator{\BH}{\mathbb{H}} % 
\DeclareMathOperator{\BK}{\mathbb{K}} % 
\DeclareMathOperator{\BL}{\mathbb{L}} % 
\DeclareMathOperator{\BO}{\mathbb{O}} % 
\DeclareMathOperator{\re}{Re}         % real part
\DeclareMathOperator{\im}{Im}         % imaginary part
\DeclareMathOperator{\Ker}{Ker}
%
\DeclareMathOperator{\meet}{\wedge}
\DeclareMathOperator{\join}{\vee}
\DeclareMathOperator{\union}{\cup}
\DeclareMathOperator{\inter}{\cap}  % intersection
\DeclareMathOperator{\iso}{\cong}   % isomorphic
\DeclareMathOperator{\piso}{\simeq} % polynomially isomorphic
\newcommand{\Star}{^ *}
\DeclareMathOperator{\compose}{{\scriptstyle \circ}} % f o g
%
\newcommand{\wec}[1]{{\mathbf{#1}}}  % notation for vectors in algebras
%\newcommand{\wec}[1]{{\bar{#1}}} % which one do you prefer?
\newcommand{\wrel}[1]{\;#1\;}     % a\wrel{\alpha}b = a \alpha b
%              % this provides a flexible way to change the spacing
%
\newcommand{\m}[1]{{\mathbf{\uppercase{#1}}}}
\newcommand{\vr}[1]{{\mathcal{\uppercase {#1}}}}
%
\DeclareMathOperator{\Var}{\mathbf{V}}
\DeclareMathOperator{\Ho}{\mathbf{H}}
\DeclareMathOperator{\Su}{\mathbf{S}}
\DeclareMathOperator{\Pd}{\mathbf{P}}
\DeclareMathOperator{\Iso}{\mathbf{I}}
\DeclareMathOperator{\Ps}{\mathbf{P}_s}
\newcommand{\fin}[1]{{\mathbf{#1}}_{\text{\upshape fin}}}
%
\DeclareMathOperator{\Pol}{Pol}
\DeclareMathOperator{\Clo}{Clo}
\DeclareMathOperator{\Con}{Con}
\DeclareMathOperator{\Conb}{\mathbf{Con}} % Con(A) as a lattice
\DeclareMathOperator{\cg}{Cg}
\DeclareMathOperator{\sg}{Sg}
\newcommand{\Cg}[1]{{\cg}^{\m{#1}}}
\newcommand{\Sg}[1]{{\sg}^{\m{#1}}}
%
\newcommand{\M}[3]{\text{\upshape M}_{\m{#1}} (#2, #3)}
\newcommand{\minl}[1]{\text{\upshape Min}_{\m{#1}}}
%
\DeclareMathOperator{\typ}{typ}
\DeclareMathOperator{\ann}{ann}
\DeclareMathOperator{\Sep}{Sep}
\DeclareMathOperator{\ssb}{ssb}
\DeclareMathOperator{\sst}{sst}
%
\DeclareMathOperator{\utyp}{\mathbf{1}}% unary      
\DeclareMathOperator{\atyp}{\mathbf{2}}% affine     
\DeclareMathOperator{\btyp}{\mathbf{3}}% boolean    
\DeclareMathOperator{\ltyp}{\mathbf{4}}% lattice    
\DeclareMathOperator{\styp}{\mathbf{5}}% semilattice
\DeclareMathOperator{\ityp}{\mathbf{i}}
\DeclareMathOperator{\jtyp}{\mathbf{j}}
%
\DeclareMathOperator{\solv}{\stackrel{s}{\sim}}
\DeclareMathOperator{\ssolv}{\stackrel{ss}{\sim}}
%
\newcommand{\bd}{\begin{description}}
\newcommand{\ed}{\end{description}}
%
\newcommand{\lb}{\langle} % <...>
\newcommand{\rb}{\rangle}
\newcommand{\lv}{{[\![}}  % [[...]]
\newcommand{\rv}{{]\!]}}
%
\newcommand{\ul}{\underline}
\newcommand{\ol}{\overline}
%
\newcommand{\vep}{\varepsilon}
\newcommand{\di}{\diamond}
%
\newcommand{\cl}{{\sf cl}}

\begin{document}

\noindent
{\large \bf Additional notes \#12.}

\bigskip

\noindent
{\bf Review of Subnormal Series and Solvable Groups.}

\bigskip

\noindent
{\bf Defns.} Let $G$ be a group. 
\begin{enumerate}
\item[(1)] A {\em factor} (or {\em divisor})
of $G$ is quotient group $H/N$ where $N\lhd H < G$.
\item[(2)] A {\em subnormal series} of $G$ is a 
finite sequence of subgroups $N_0<\cdots< N_k$
with $N_i\lhd N_{i+1}$ for each $i$.
The factors of this series are the groups $N_{i+1}/N_i$.
\item[(3)] A {\em refinement} of a subnormal series
$N_0\lhd \cdots\lhd N_k$ is a subnormal series
that includes all $N_i$ together with possibly
more intermediate groups. 
\item[(4)] A {\em composition series} is a subnormal series
from $N_0 = \{1\}$ to $N_k=G$ that has no proper refinement. 
Its factors are called {\em composition factors}.
\end{enumerate}

\medskip

\noindent
{\bf Thm.} (Behavior of subnormal series.)
\begin{enumerate}
\item[(1)] If 
$N_0\lhd N_1\lhd\cdots\lhd N_k$ is a subnormal series
and $1\leq i\leq j\leq n$, 
then $N_i\lhd N_{i+1}\lhd\cdots\lhd N_j$  is another
subnormal series.
Its factors are among the factors of the
original series.
\item[(2)] If $N_0\lhd \cdots\lhd N_k$ 
and $N_k\lhd\cdots\lhd N_{\ell}$ are subnormal series,
then $N_0\lhd \cdots\lhd N_{\ell}$  is one.
Its factors are the union of the factors of the
two original series.
\item[(3)] If $N_0\lhd \cdots\lhd N_k$ 
is a subnormal series, then the factors
of any refinement of this series are among the
subgroups and quotients of the factors of this series.
\item[(4)] If 
$N_0\lhd N_1\lhd\cdots\lhd N_k$ is a subnormal series
and $H$ is any subgroup, then 
$H\cap N_0\lhd H\cap N_1\lhd\cdots\lhd H\cap N_k$ 
is a subnormal series. The factor 
$(H\cap N_{i+1})/(H\cap N_i)$ of this series is isomorphic
to a subgroup of the corresponding factor $N_{i+1}/N_i$
of the original series.
\item[(5)] If 
$N_0\lhd N_1\lhd\cdots\lhd N_k$ is a subnormal series
and $H$ is a {\bf normal} subgroup, then 
$H\vee N_0\lhd H\vee N_1\lhd\cdots\lhd H\vee N_k$ 
is a subnormal series. The factor 
$(H\vee N_{i+1})/(H\vee N_i)$ of this series is 
a quotient of the corresponding factor $N_{i+1}/N_i$
of the original series.
\end{enumerate}

\medskip

Let $\vr C$ be a class of groups. We will say that a subnormal
series {\em has type ${\vr C}$} is all factors of the series
isomorphic to groups in $\vr C$.

\medskip

\noindent
{\bf Thm.}
Let $\vr C$ be a class of groups closed under the formation
of subgroups and quotient groups. Let $G$ be a finite group.
The following conditions are equivalent.
\begin{enumerate}
\item[(1)] $G$ has a set of subnormal series
$N_0^j\lhd\cdots \lhd N_k^j=G$, $j=1,\ldots,n$, 
of type $\vr C$ such that 
$\bigcap_{j=1}^n N_0^j = \{1\}$.
\item[(2)] $G$ has a subnormal series 
$\{1\} = N_0\lhd\cdots \lhd N_k=G$ of type $\vr C$.
\item[(3)] $G$ has a composition series of type $\vr C$.
\item[(4)] All simple factors of $G$ are in $\vr C$.
\item[(5)] All composition series of $G$ have type $\vr C$.
\end{enumerate}

\bigskip

\noindent
{\bf Defn.} A group is {\em solvable} if it has
a subnormal series 
$\{1\} = N_0\lhd\cdots \lhd N_k=G$ 
with abelian factors.

\medskip

\noindent
{\bf Cor.} A finite group is solvable if and only if
its composition factors are cyclic of prime order.

\medskip

\noindent
{\bf Cor.} If $H\lhd G$, then $G$ is solvable if and only
if $H$ and $G/H$ are solvable.

\bigskip

\noindent
{\bf Defn.} An extension ${\BF}<{\BE}$ is
{\em solvable} if it is Galois and
its Galois group is solvable.

\medskip

\noindent
{\bf Thm.} A Galois extension ${\BF}<{\BE}$
is solvable if and only if there is a
series ${\BF}={\BK}_k<\cdots<{\BK}_0 = {\BE}$
of Galois extensions of prime order.

\medskip

\noindent
{\bf Thm.} 
\begin{enumerate}
\item[(1)] The composite of two 
solvable (finite) extensions of ${\BF}$
is solvable.
\item[(2)] If ${\BF}={\BK}_k<\cdots<{\BK}_0 = {\BE}$
is a series of (finite) 
solvable extensions, then the normal closure
of ${\BE}$ over ${\BF}$ is a solvable extension of 
${\BF}$.
\end{enumerate}

\end{document}

