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% Most of the following are not required for the majority
% of cheat sheets but are needed for some symbol support.
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% Document Info
\author{dimples}
\pdfinfo{
  /Title (abstract-algebra.pdf)
  /Creator (Cheatography)
  /Author (dimples)
  /Subject (Abstract Algebra Cheat Sheet)
}

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\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{C}
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    \vspace{-7pt}
    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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\columnbreak
\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Abstract Algebra Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{dimples} via \textcolor{DarkBackground}{\uline{cheatography.com/207890/cs/44440/}}}
\end{tabulary}
\end{multicols}}

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\noindent
\begin{multicols}{3}
\begin{tabulary}{5.8cm}{LL}
  \SetRowColor{FootBackground}
  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}dimples \\
  \uline{cheatography.com/dimples} \\
  \end{tabulary}
\vfill
\columnbreak
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  \SetRowColor{FootBackground}
  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 24th September, 2024.\\
   Page {\thepage} of \pageref{LastPage}.
\end{tabulary}
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  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\begin{document}
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% www.emerson.emory.edu/services/latex/latex_169.html
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{x{2.23965 cm} x{2.73735 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Definition of Groups}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Binary Operation}} & Let G be a set. A binary operation on G is a function that assigns each ordered pair of elements of G an element of G. \tn 
% Row Count 6 (+ 6)
% Row 1
\SetRowColor{white}
{\bf{Group}} & Let G be a set together with a binary operation (usually called multiplication) that assigns to each ordered pair (a, b) of elements of G an element in G denoted by ab. We say G is a group under this operation if the following three properties are satisfied. \tn 
% Row Count 18 (+ 12)
% Row 2
\SetRowColor{LightBackground}
{\bf{Properties to Satisfy (Group)}} & 1. Closure 2. Associativity 3. Identity 4. Inverses \tn 
% Row Count 21 (+ 3)
% Row 3
\SetRowColor{white}
{\bf{Abelian group}} & If a group has the property that ab = ba for every pair of elements a and b, we say the group is Abelian. \tn 
% Row Count 26 (+ 5)
% Row 4
\SetRowColor{LightBackground}
{\bf{Associativity}} & The operation is associative; that is, (ab)c = a(bc) for all a, b, c in G. \tn 
% Row Count 30 (+ 4)
\end{tabularx}
\par\addvspace{1.3em}

\vfill
\columnbreak
\begin{tabularx}{5.377cm}{x{2.23965 cm} x{2.73735 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Definition of Groups (cont)}}  \tn
% Row 5
\SetRowColor{LightBackground}
{\bf{Identity}} & There is an element e (called the identity) in G such that ae = ea = a for all a in G. \tn 
% Row Count 4 (+ 4)
% Row 6
\SetRowColor{white}
{\bf{Inverses}} & For each element a in G, there is an element b in G (called an inverse of a) such that ab = ba = e. \tn 
% Row Count 9 (+ 5)
% Row 7
\SetRowColor{LightBackground}
{\bf{Modular Arithmetic}} & When a = qn + r, where q is the quotient and r is the remainder upon dividing a by n, we write a mod n = r. \tn 
% Row Count 14 (+ 5)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Elementary Properties of Groups}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Theorem 2.1 Uniqueness of the Identity}} & In a group G, there is only one identity element. \tn 
% Row Count 3 (+ 3)
% Row 1
\SetRowColor{white}
{\bf{Theorem 2.2 Cancellation}} & In a group G, the right and left cancellation laws hold; that is, ba = ca implies b = c, and ab = ac implies b = c. \tn 
% Row Count 9 (+ 6)
% Row 2
\SetRowColor{LightBackground}
{\bf{Theorem 2.3 Uniqueness of Inverses}} & For each element a in a group G, there is a unique element b in G such that ab = ba = e. \tn 
% Row Count 14 (+ 5)
% Row 3
\SetRowColor{white}
{\bf{Theorem 2.4 Socks-Shoes Property}} & For group elements a and b, (ab)\textasciicircum{}-1\textasciicircum{} = b\textasciicircum{}-1\textasciicircum{}a\textasciicircum{}-1\textasciicircum{}. \tn 
% Row Count 17 (+ 3)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.33919 cm} x{2.63781 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Subgroup Tests}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{One-Step Subgroup Test}} & Let G be a group and H a nonempty subset of G. If ab\textasciicircum{}-1\textasciicircum{} is in H whenever a and b are in H, then H is a subgroup of G. (In additive notation, if a - b is in H whenever a and b are in H, then H is a subgroup of G.) \tn 
% Row Count 11 (+ 11)
% Row 1
\SetRowColor{white}
{\bf{Two-Step Subgroup Test}} & Let G be a group and let H be a nonempty subset of G. If ab is in H whenever a and b are in H (H is closed under the operation), and a\textasciicircum{}-1\textasciicircum{} is in H whenever a is in H (H is closed under taking inverses), then H is a subgroup of G. \tn 
% Row Count 22 (+ 11)
% Row 2
\SetRowColor{LightBackground}
{\bf{Theorem 3.3 Finite Subgroup Test}} & Let H be a nonempty finite subset of a group G. If H is closed under the operation of G, then H is a subgroup of G. \tn 
% Row Count 28 (+ 6)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{2.33919 cm} x{2.63781 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Examples of Subgroups}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Theorem 3.4 \textless{}a\textgreater{} Is a Subgroup}} & Let G be a group, and let a be any element of G. Then, \textless{}a\textgreater{} is a subgroup of G. \tn 
% Row Count 4 (+ 4)
% Row 1
\SetRowColor{white}
{\bf{Center of a Group}} & The center, Z(G), of a group G is the subset of elements in G that commute with every element of G. In symbols, \{\{nl\}\}  {\bf{Z(G) = \{a ∈ G | ax = xa for all x in G\}}}. \tn 
% Row Count 12 (+ 8)
% Row 2
\SetRowColor{LightBackground}
{\bf{Theorem 3.5 Center Is a Subgroup}} & The center of a group G is a subgroup of G. \tn 
% Row Count 15 (+ 3)
% Row 3
\SetRowColor{white}
{\bf{Centralizer of a in G}} & Let a be a fixed element of a group G. The centralizer of a in G, C(a), is the set of all elements in G that commute with a. In symbols, \{\{nl\}\}   {\bf{C(a) = \{g ∈ G | ga = ag\}}}. \tn 
% Row Count 24 (+ 9)
% Row 4
\SetRowColor{LightBackground}
{\bf{Theorem 3.6 C(a) Is a Subgroup}} & For each a in a group G, the centralizer of a is a subgroup of G. \tn 
% Row Count 28 (+ 4)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}

\begin{tabularx}{5.377cm}{x{1.84149 cm} x{3.13551 cm} }
\SetRowColor{DarkBackground}
\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Terminology and Notation}}  \tn
% Row 0
\SetRowColor{LightBackground}
{\bf{Order of a Group}} & The number of elements of a group (finite or infinite) is called its order. We will use |G| to denote the order of G. \tn 
% Row Count 5 (+ 5)
% Row 1
\SetRowColor{white}
{\bf{Order of an Element}} & The order of an element g in a group G is the smallest positive integer n such that g\textasciicircum{}n\textasciicircum{} = e. (In additive notation, this would be ng = 0.) If no such integer exists, we say that g has infinite order. The order of an element g is denoted by |g|. \tn 
% Row Count 15 (+ 10)
% Row 2
\SetRowColor{LightBackground}
{\bf{Subgroup}} & If a subset H of a group G is itself a group under the operation of G, we say that H is a subgroup of G. \tn 
% Row Count 20 (+ 5)
\hhline{>{\arrayrulecolor{DarkBackground}}--}
\end{tabularx}
\par\addvspace{1.3em}


% That's all folks
\end{multicols*}

\end{document}