Abelian groups

From Structures

\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}