```
% ===============================================
% MATH 3053: Abstract Algebra I Fall 2017
% hw_revised.tex
% Template for revised homework submission
% ===============================================
% READ THE FOLLOWING CAREFULLY!!!
% ===============================================
% When you produce a PDF version of this document
% to turn in, change the filename to hwX-name.pdf
% replacing X with the homework assignment number
% and name with your last name.
% ===============================================
% -----------------------------------------------
% The preamble that follows can be ignored. Go on
% down to the section that says "START HERE"
% -----------------------------------------------
\documentclass{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amsthm,amssymb}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\C}{\mathbb{C}}
\newenvironment{theorem}[2][Theorem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{lemma}[2][Lemma]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{exercise}[2][Exercise]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{problem}[2][Problem]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{question}[2][Question]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{corollary}[2][Corollary]{\begin{trivlist}
\item[\hskip \labelsep {\bfseries #1}\hskip \labelsep {\bfseries #2.}]}{\end{trivlist}}
\newenvironment{solution}{\begin{proof}[Solution]}{\end{proof}}
\begin{document}
% ------------------------------------------ %
% START HERE %
% ------------------------------------------ %
\title{Homework X Revised Draft} % Replace X with the appropriate number
\author{Author's Name\\Abstract Algebra I} % Replace "Author's Name" with your name
\maketitle
% -----------------------------------------------------
% The following two environments (theorem, proof) are
% where you will enter the statement and proof of your
% first problem for this assignment.
%
% In the theorem environment, you can replace the word
% "theorem" in the \begin and \end commands with
% "exercise", "problem", "lemma", etc., depending on
% what you are submitting. Replace the "x.yz" with the
% appropriate number for your problem.
%
% If your problem does not involve a formal proof, you
% can change the word "proof" in the \begin and \end
% commands with "solution".
% -----------------------------------------------------
\begin{theorem}{x.yz}
Replace this text with the statement of the theorem you are proving.
\end{theorem}
\begin{proof}
Replace this text with the details of your proof or solution.
\end{proof}
% -----------------------------------------------------
% Second problem
% -----------------------------------------------------
\vspace{0.25in} % This just adds some space between problems 1 and 2.
\begin{theorem}{x.yz}
Replace this text with the statement of the theorem you are proving.
\end{theorem}
\begin{proof}
Replace this text with the details of your proof or solution.
\end{proof}
% -----------------------------------------------------
% Third problem
% -----------------------------------------------------
\vspace{0.25in} % This adds some space between problems 2 and 3.
\begin{theorem}{x.yz}
Replace this text with the statement of the theorem you are proving.
\end{theorem}
\begin{proof}
Replace this text with the details of your proof or solution.
\end{proof}
% -----------------------------------------------
% Ignore everything that appears below this.
% -----------------------------------------------
\end{document}
```