newtheorem , newenvironment Modifikationen |
09.07.2011, 20:02 | Jeremy124 | Auf diesen Beitrag antworten » |
newtheorem , newenvironment Modifikationen an einer Umgebung, die ich gerne benutzen möchte. Es klappt schon ganz gut. Nur möchte ich nach den Titeln einen Punkt haben. Also: Lemma 1. Theorem 3. Auch möchte ich für die Definitionen einen Punkt, aber davor noch eine Nummerierung. Danke für eure Hilfe _________________________________________________________________ Aus einer von TeXworks gegebener Vorlage, ist bisher folgendes entstanden: \documentclass[12pt]{article} \usepackage{amssymb} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{corollary}{Corollary} \newtheorem{proposition}{Proposition} \newenvironment{proof}[1][Proof]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}} \newenvironment{definition}[1][Definition]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}} \newenvironment{example}[1][Example]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}} \newenvironment{remark}[1][Remark]{\begin{trivlist} \item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}} \newcommand{\qed}{\hfill \ensuremath{\Box}} \title{Brief Article} \author{The Author} %\date{} % Activate to display a given date or no date (if empty), % otherwise the current date is printed \begin{document} \maketitle \section{First section} \begin{definition} Let $H$ be a subgroup of a group~$G$. A \emph{left coset} of $H$ in $G$ is a subset of $G$ that is of the form $xH$, where $x \in G$ and $xH = \{ xh : h \in H \}$. Similarly a \emph{right coset} of $H$ in $G$ is a subset of $G$ that is of the form $Hx$, where $Hx = \{ hx : h \in H \}$ \end{definition} Note that a subgroup~$H$ of a group $G$ is itself a left coset of $H$ in $G$. \begin{lemma} \label{LeftCosetsDisjoint} Let $H$ be a subgroup of a group $G$, and let $x$ and $y$ be elements of $G$. Suppose that $xH \cap yH$ is non-empty. Then $xH = yH$. \end{lemma} \begin{proof}[Proof of Lemma 1] Let $z$ be some element of $xH \cap yH$. Then $z = xa$ for some $a \in H$, and $z = yb$ for some $b \in H$. If $h$ is any element of $H$ then $ah \in H$ and $a^{-1}h \in H$, since $H$ is a subgroup of $G$. But $zh = x(ah)$ and $xh = z(a^{-1}h)$ for all $h \in H$. Therefore $zH \subset xH$ and $xH \subset zH$, and thus $xH = zH$. Similarly $yH = zH$, and thus $xH = yH$, as required. \qed \end{proof} \begin{lemma} \label{SizeOfLeftCoset} Let $H$ be a finite subgroup of a group $G$. Then each left coset of $H$ in $G$ has the same number of elements as $H$. \end{lemma} \begin{proof} Let $H = \{ h_1, h_2,\ldots, h_m\}$, where $h_1, h_2,\ldots, h_m$ are distinct, and let $x$ be an element of $G$. Then the left coset $xH$ consists of the elements $x h_j$ for $j = 1,2,\ldots,m$. Suppose that $j$ and $k$ are integers between $1$ and $m$ for which $x h_j = x h_k$. Then $h_j = x^{-1} (x h_j) = x^{-1} (x h_k) = h_k$, and thus $j = k$, since $h_1, h_2,\ldots, h_m$ are distinct. It follows that the elements $x h_1, x h_2,\ldots, x h_m$ are distinct. We conclude that the subgroup~$H$ and the left coset $xH$ both have $m$ elements, as required.\qed \end{proof} \begin{theorem} \emph{(Lagrange's Theorem)} \label{Lagrange} Let $G$ be a finite group, and let $H$ be a subgroup of $G$. Then the order of $H$ divides the order of $G$. \end{theorem} \begin{proof} Each element~$x$ of $G$ belongs to at least one left coset of $H$ in $G$ (namely the coset $xH$), and no element can belong to two distinct left cosets of $H$ in $G$ (see Lemma~\ref{LeftCosetsDisjoint}). Therefore every element of $G$ belongs to exactly one left coset of $H$. Moreover each left coset of $H$ contains $|H|$ elements (Lemma~\ref{SizeOfLeftCoset}). Therefore $|G| = n |H|$, where $n$ is the number of left cosets of $H$ in $G$. The result follows.\qed \end{proof} \begin{remark} Preceding proof is not trivial. \end{remark} \end{document} |
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