我正在做课堂笔记,并决定为我的代数部分列出定理和定义会很有用。定理环境本身运行良好,但当我使用包\listoftheorems
中的命令时thmtools
,会出现一个带有标题的页面,但上面没有条目。
我正在使用 TexMaker。我的代码如下。
\documentclass[oneside]{book}
\usepackage[utf8]{inputenc}
\usepackage{amsthm,amsmath,amssymb,amsfonts}
\usepackage{graphicx}
\usepackage[svgnames]{xcolor}
\usepackage{thmtools}
\usepackage{subfiles}
\usepackage{indentfirst}
\usepackage{array}
\usepackage[hidelinks]{hyperref}
\usepackage[left=1in,right=1in,top=1in,bottom=1in, bindingoffset=0in]{geometry}
\setcounter{tocdepth}{0}
\graphicspath{{Images/}{../Images/}}
\declaretheorem{theorem}
\declaretheorem{example}
\declaretheorem{definition}
\renewcommand{\listtheoremname}{List of Theorems and Definitions}
\begin{document}
\listoftheorems[ignoreall, show={theorem,definition}]
\chapter{Group Theory}
\section{30,000ft View}
Group theory is the theory of reversible action. More formally, it is the theory of symmetry and transformation.
\begin{example}
Suppose that you have a square,and wish to perform. There are a couple different flips one could do, vertically or horizontally. We could name these as function f,g, which act on the vertex set $V=\{A,B,C,D\}$ like so:
$$f:V\rightarrow V \text{defined by}$$ $$f:A\mapsto B, f:B\mapsto A, f:D\mapsto C, \text{and } f:C\mapsto D$$
$$g:V\rightarrow V \text{defined by}$$ $$g:A\mapsto D, g:D\mapsto A, g:B\mapsto C, \text{and } g:C\mapsto B$$
or the identity function 1, which corresponds to doing nothing.
$$1:V\rightarrow V \text{defined by}$$
$$1:A\mapsto A, 1:B\mapsto B, 1:C\mapsto C,\text{and } 1:D\mapsto D$$.
Since these are functions, we may compose them. In this case, this corresponds to multiple flips. The group G is (informally) the set of all things I can do the the vertex set under these functions.
$$G=\{1,f,g,g\circ f, f\circ g, f\circ g\circ f,...\}$$
This particular group is not actually infinite, because $f=f^{-1}$, $g=g^{-1}$, $f\circ 1=1\circ f= f$ and$g\circ 1=1\circ g= g$. The full group is then just
$$G=\{1,f,g,f\circ g\}$$
and we can prove this with the multiplication table.
\begin{example}
Suppose we have two binary switches. Define l=flip the left switch, and r=flip the right switch, and 1=do nothing. These are function of the set of states $S=\{UU,UD,DU,DD\}$. Thus, the group is
$$H=\{1,l,r,lr\}$$ This is the entire group for the same reasons as in the previous example.
\end{example}
Since H and G have the same multiplication table, up to re-labeling, we say these groups are isomorphic.
Both of these examples were groups of functions, and these were all bijective functions. In the context of group theory, this is called a permutation.
\begin{definition}
Suppose $S$ is a set. Then $f:S\rightarrow S$ is a permutation iff $f$ is a bijection.
\end{definition}
\begin{definition}
For $n\in \mathbb{Z}$, $S_n$ is the set of permutations of the set $\{1,...,n\}$.
\end{definition}
Summary: With $S_n$, we have an associative binary operation such that there exists a two-sided identity, and two sided inverses. This makes it a group.
\end{document}