Abelian groups

http://math.chapman.edu/structuresold/files/Abelian_groups.pdf

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\begin{document}
\textbf{\Large Abelian groups}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Abelian_groups}{edit}

\abbreviation{AbGrp}

\begin{definition}
An \emph{abelian group} is a structure $\mathbf{G}=\langle
G,+,-,0\rangle$, where $+$ is an infix binary operation, called the 
\emph{group addition}, $-$ is a prefix unary operation, called the 
\emph{group negative} and $0$ is a constant (nullary operation), called the \emph{additive identity element}, such that

$+$ is commutative:  $x+y=y+x$

$+$ is associative:  $(x+y)+z=x+(y+z)$

$0$ is an additive identity for $+$:  $0+x=x$

$-$ gives an additive inverse for $+$:  $-x+x=0$
\end{definition}

\begin{morphisms}
Let $\mathbf{G}$ and $\mathbf{H}$ be abelian groups. A morphism from $\mathbf{G}$ to $\mathbf{H}$ is a function $h:G\rightarrow H$ that is a
homomorphism: 
$h(x+y)=h(x)+h(y)$

Remark: It follows that $h(-x)= -h(x)$, $h(0)=0$.

\end{morphisms}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
$\langle \mathbb{Z}, +, -, 0\rangle $, the integers, with addition, unary subtraction, and zero. The variety of abelian groups is generated by this algebra.
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & variety\\\hline
Equational theory & decidable in polynomial time\\\hline
Quasiequational theory & decidable\\\hline
First-order theory & decidable \cite{Szmielew1949}\\\hline
Locally finite & no\\\hline
Residual size & $\omega$\\\hline
Congruence distributive & no ($\mathbb{Z}_{2}\times \mathbb{Z}_{2}$)\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$, $p(x,y,z)=x-y+z$\\\hline
Congruence regular & yes, congruences are determined by subalgebras\\\hline
Congruence uniform & yes\\\hline
Congruence types & permutational\\\hline
Congruence extension property & yes, if $K\le H\le G$ then $K\le G$\\\hline
Definable principal congruences & no\\\hline
Equationally def. pr. cong. & no\\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & yes\\\hline
Epimorphisms are surjective & yes\\\hline
\end{tabular}
\end{properties}
\end{table}

\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr}
f(1)= &1\\
f(2)= &1\\
f(3)= &1\\
f(4)= &2\\
f(5)= &1\\
\end{array}$\qquad
$\begin{array}{lr}
f(6)= &1\\
f(7)= &1\\
f(8)= &3\\
f(9)= &2\\
f(10)= &1\\
\end{array}$\qquad
$\begin{array}{lr}
f(11)= &1\\
f(12)= &2\\
f(13)= &1\\
f(14)= &1\\
f(15)= &1\\
\end{array}$

\url{http://www.research.att.com/projects/OEIS?Anum=A000688}
\end{finite_members}

\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\ 

\href{Boolean_groups.pdf}{Boolean groups} 

\href{Commutative_rings.pdf}{Commutative rings} 

\end{subclasses}

\begin{superclasses}\ 

\href{Groups.pdf}{Groups} 

\href{Commutative_monoids.pdf}{Commutative monoids} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Szmielew1949}
W. Szmielew, \emph{Decision problem in group theory},
Library of the Tenth International Congress of Philosophy, 
Amsterdam, August 11--18, 1948, Vol.1, Proceedings of the Congress,
1949, 763--766 \href{"http://www.ams.org/mathscinet-getitem?mr=10:500a"}{MRreview}

\end{thebibliography}

\end{document}
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