Csdrhrt

I am trying to remove the excessive space between a definition.

documentclass[11pt, a4paper]{article}

usepackage{eurosym}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

usepackage{amsmath}

usepackage{amsfonts}

usepackage{bm}

usepackage{amsfonts, graphicx, verbatim, amsmath,amssymb}

usepackage{color}

usepackage{lipsum}

usepackage{array}

usepackage{setspace}



setcounter{MaxMatrixCols}{10}

%TCIDATA{OutputFilter=Latex.dll}

%TCIDATA{Version=5.50.0.2953}

%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}

%TCIDATA{BibliographyScheme=Manual}

%TCIDATA{LastRevised=Sunday, November 26, 2017 16:01:29}

%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}



setlength{textheight}{22cm}setlength{textwidth}{16cm}

setlength{topmargin}{-1.5cm}

setlength{oddsidemargin}{-0.5cm}setlength{evensidemargin}{-0.5cm}

providecommand{U}[1]{protectrule{.1in}{.1in}}

setlength{textheight}{24cm}setlength{textwidth}{16.5cm}

setlength{topmargin}{-1.5cm}

setlength{oddsidemargin}{0.5cm}setlength{evensidemargin}{0.5cm}

newtheorem{theorem}{Theorem}[section]

newtheorem{acknowledgement}[theorem]{Acknowledgement}

newtheorem{algorithm}[theorem]{Algorithm}

newtheorem{axiom}[theorem]{Axiom}

newtheorem{case}[theorem]{Case}

newtheorem{claim}[theorem]{Claim}

newtheorem{Theorem}[theorem]{Theorem}

newtheorem{conclusion}[theorem]{Conclusion}

newtheorem{condition}[theorem]{Condition}

newtheorem{conjecture}[theorem]{Conjecture}

newtheorem{corollary}[theorem]{Corollary}

newtheorem{corol}[theorem]{Corollary}

newtheorem{Fact}[theorem]{Fact}

newtheorem{Corollary}[theorem]{Corollary}

newtheorem{criterion}[theorem]{Criterion}

newtheorem{definition}[theorem]{Definition}

newtheorem{Definition}[theorem]{Definition}

newtheorem{example}[theorem]{Example}

newtheorem{exercise}[theorem]{Exercise}

newtheorem{lemma}[theorem]{Lemma}

newtheorem{Lemma}[theorem]{Lemma}

newtheorem{fact}[theorem]{Fact}

newtheorem{lma}[theorem]{Lemma}

newtheorem{notation}[theorem]{Notation}

newtheorem{problem}[theorem]{Problem}

newtheorem{proposition}[theorem]{Proposition}

newtheorem{prop}[theorem]{Proposition}

newtheorem{Property}[theorem]{Property}

newtheorem{property}[theorem]{Property}

newtheorem{remark}[theorem]{Remark}

newtheorem{Comment}[theorem]{Comment}

newtheorem{solution}[theorem]{Solution}

newtheorem{summary}[theorem]{Summary}

newenvironment{proof}[1][Proof]{textbf{#1.} }{ rule{0.5em}{0.5em}}

newcommand{ve}{varepsilon}

newcommand{cvgpr}{xrightarrow{text{upshapetiny P}}}

newcommand{cvgdist}{xrightarrow{mathrm{d}}}

newcommand{G}{{mathcal{G}}}

newcommand{Kx}{{cal K}}

newcommand{tod}{to^{cal D}}

newcommand{ls}{limsup_{ntoinfty}}

newcommand{rE}{mathbb{E}}

newcommand{A}{{mathcal{A}}}

newcommand{rP}{mathbb{P}}

newcommand{p}{{mathbb{P}}}

newcommand{Z}{{mathbb{Z}}}

newcommand{Be}{{rm Be}}

newcommand{re}{mathrm{e}}

newcommand{ep}{varepsilon}

newcommand{Bin}{{rm Bin}}

newcommand{qand}{quadmbox{and}quad}

newcommand{quso}{quadmbox{so}quad}

newcommand{Nn}{{bf N}}

newcommand{St}{underline{rm S}}

newcommand{Rt}{underline{rm R}}

newcommand{It}{underline{rm I}}

newcommand{one}{{bf 1}}

newcommand{Ups}{{Upsilon}}

newcommand{iu}{{imkern1mu}}

newcommand{II}{{mathcal{I}}}

newcommand{Var}{{rm Var}}

newcommand{var}{{rm Var}}

newcommand{Cov}{{rm cov}}

newcommand{cov}{{rm cov}}

newcommand{corr}{{rm corr}}

newcommand{lhs}{{rm lhs}}

newcommand{rhs}{{rm rhs}}

newcommand{ra}{rightarrow}

newcommand{I}{{mathbf 1}}

newcommand{R}{{mathbb R}}

newcommand{N}{{mathbb N}}

newcommand{LL}{{mathbb L}}

newcommand{E}{{mathbb{E}}}

newcommand{bin}{{rm Bin}}

newcommand{Pois}{{rm Pois}}

newcommand{Po}{{rm Pois}}

newcommand{Bi}{{cal B}}

newcommand{ri}{mathrm{i}}

newcommand{rd}{mathrm{d}}

newcommand{XXi}{Xi_{k,m}^{(n)}}

newcommand{xxi}{bar{xi}}

newcommand{qedhere}{{diamond}}

newcommand{eqdef}{stackrel{mathrm{def}}{=}}

newcommand{eqdist}{stackrel{mathrm{D}}{=}}

newcommand{braket}[2]{{langle{#1|#2}rangle}}

newcommand{independent}{perp}

newcommand{bb}{begin{eqnarray*}}

newcommand{ee}{end{eqnarray*}}

newcommand{bbb}{begin{eqnarray}}

newcommand{eee}{end{eqnarray}}

newcommand{F}{{mathcal{F}}}

newcommand{qed}{$diamond$}

parindent 0pt

setlength{parindent}{0pt}

%newcommand{forceindent}{leavevmode{parindent=3emindent



begin{document}

section{Introduction}

We shall go over some basic knowledge hence we begin by giving some definitions. 

begin{definition} 

A group is a set $G$ together with a binary operation $*$ on $G$ satisfying

the following properties:\

doublespacing{(G1) Closure:     $forall x,y in G, x * y in G.$\

(G2) Associativity:  $forall x,y, z in G, (x * y) * z = x * (y * z).$\

(G3) Identity:   There is an element $e in G$ such that $e * x = x * e = x$ for all $x in G.$\

(G4) Inverses:   For any $x in G$ there is an element $y in G$ such that $x * y = y * x = e.$\}

end{definition}

end{document}

now I wish to remove the 'extra' space where it says "properties" by moving it one line up and keeping the rest the same

EDIT: I wish to add the following:

begin{definition}

A group $G$ is called an abelian group if the following axiom is satisfied:\



(G5) Commutativity: $forall x,y in G, x * y = y * x.$

end{definition}

You should first get rid of most code in the preamble and define only the commands and structures that you really need.

By the way, commands such as rm and cal have been deprecated for twenty years.

Don't use \ for ending paragraphs in standard text. A wider spacing for the items in the definition is better obtained by using enumerate rather than doublespacing (which is not a command taking an argument).

documentclass[11pt, a4paper]{article}

usepackage{amsmath,amsthm,enumitem}



newtheorem{theorem}{Theorem}[section]



theoremstyle{definition}

newtheorem{definition}[theorem]{Definition}



begin{document}



section{Introduction}



We shall go over some basic knowledge hence we begin by giving some definitions. 



begin{definition} 

A group is a set $G$ together with a binary operation $*$ on $G$ satisfying

the following properties:

begin{enumerate}[label=(Garabic*)]

item Closure: $forall x,y in G, x * y in G$.

item Associativity: $forall x,y, z in G, (x * y) * z = x * (y * z)$.

item Identity: There is an element $e in G$ such that $e * x = x * e = x$ for all $x in G$.

item Inverses: For any $x in G$ there is an element $y in G$ such that $x * y = y * x = e$.

end{enumerate}

end{definition}



end{document}

The enumitem package has several features, for instance series and resume:

documentclass[11pt, a4paper]{article}

usepackage{amsmath,amsthm,enumitem}



newtheorem{theorem}{Theorem}[section]



theoremstyle{definition}

newtheorem{definition}[theorem]{Definition}



begin{document}



section{Introduction}



We shall go over some basic knowledge hence we begin by giving some definitions. 



begin{definition} 

A group is a set $G$ together with a binary operation $*$ on $G$ satisfying

the following properties:

begin{enumerate}[label=(Garabic*),series=group]

item Closure: $forall x,y in G, x * y in G$.

item Associativity: $forall x,y, z in G, (x * y) * z = x * (y * z)$.

item Identity: There is an element $e in G$ such that $e * x = x * e = x$ for all $x in G$.

item Inverses: For any $x in G$ there is an element $y in G$ such that $x * y = y * x = e$.

end{enumerate}

end{definition}



begin{definition}

A group $G$ is called an abelian group if the following axiom is satisfied:

begin{enumerate}[label=(Garabic*),resume=group]

item Commutativity: $forall x,y in G, x * y = y * x$.

end{enumerate}

end{definition}



end{document}