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ref.tex
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ref.tex
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\documentclass{amsart}
\usepackage{xspace}
\newcommand{\Reals}{\ensuremath{\mathbb{R}}\xspace}
\begin{document}
This is a reference of notation and definitions that I developed as I
often forget some of it while doing homework, and the index doesn't
have a ``notation'' index.
\begin{tabular}{lp{4in}}
$\Reals$ & When used to refer to a group means the additive group of
real numbers: $\langle \Reals, + \rangle$. (p.~26) \\
$\Reals^*$ & When used to refer to a group means the multiplicative
group of real numbers: $\langle \Reals, * \rangle$. (p.~26) \\
$\mathbb{F}(\Reals)$ & Represents the set of all functions from
\Reals to \Reals. When used to refer to a group it refers to the
additive group of functions. (p.~45) \\
$S_A$ & For any set $A$, the group of all the permutations of $A$
is called the \emph{symmetric group on} $A$, and it is represented
by the symbol $S_A$. (p.~71) See definitions below for implied operatioin.\\
$S_n$ & For any positive integer $n$, the symmetric group on the
set $\{1, 2, 3, \ldots, n \}$ is called the \emph{symmetric group
on} $n$ \emph{elements}, and is denoted by $S_n$. (p.~71) \\
\end{tabular}
Here are some definitions and some related facts.
\begin{description}
\item[permutation] By a \emph{permutation} of set $A$ we mean a
\emph{bijective function} from $A$ to $A$.
\item[symmetric group] A group of all permutations of a set. The set
may be defined by a symbol or a number. See p.~71. The operation
implied in a symmetric group is \emph{composition}, $\circ$.
\end{description}
\end{document}