Mathematical Structures: Groups

# Groups

http://mathcs.chapman.edu/structuresold/files/Groups.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Groups}

\abbreviation{Grp}

\begin{definition}
A \emph{group} is a structure $\mathbf{G}=\left\langle G,\cdot ,^{-1},e\right\rangle$, where $\cdot$ is an infix binary operation, called
the \emph{group product}, $^{-1}$ is a postfix unary operation, called the
\emph{group inverse} and $e$ is a constant (nullary operation), called the
\emph{identity element}, such that

$\cdot$ is associative: $(xy)z=x(yz)$

$e$ is a left-identity for $\cdot$: $ex=x$

$^{-1}$ gives a left-inverse: $x^{-1}x=e$.

Remark:
It follows that $e$ is a right-identity and that $^{-1}$gives a right
inverse: $xe=x$, $xx^{-1}=e$.

This definition shows that groups form a variety.
\end{definition}

\begin{morphisms}
Let $\mathbf{G}$ and $\mathbf{H}$ be groups. A morphism from $\mathbf{G}$ to
$\mathbf{H}$ is a function $h:G\rightarrow H$ that is a homomorphism:

$h(xy)=h(x)h(y)$, $h(x^{-1})=h(x)^{-1}$, $h(e)=e$
\end{morphisms}

\begin{basic_results}

\href{http://math.chapman.edu/cgi-bin/structures?Groups_otter}{Groups (otter)}
\end{basic_results}

\begin{examples}
\begin{example}
$\left\langle S_{X},\circ ,^{-1},id_{X}\right\rangle$, the collection of
permutations of a sets $X$, with composition, inverse, and identity map.
\end{example}

\begin{example}
The general linear group $\left\langle GL_{n}(V),\cdot ,^{-1},I_{n}\right\rangle$, the collection of invertible $n\times n$
matrices over a vector space $V$, with matrix multiplication, inverse, and
identity matrix.
\end{example}

\begin{example}
\href{http://math.chapman.edu/cgi-bin/structures?Groups_mace}{Groups (mace)}
\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 & undecidable\\\hline
First-order theory & undecidable\\\hline
Congruence distributive & no ($\mathbb{Z}_{2}\times \mathbb{Z}_{2}$)\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, n=2, $p(x,y,z)=xy^{-1}z$ is a Mal'cev term\\\hline
Congruence regular & yes\\\hline
Congruence uniform & yes\\\hline
Congruence types & 1=permutational\\\hline
Congruence extension property & no, consider a non-simple subgroup of a simple group\\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & no\\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & yes\\\hline
Epimorphisms are surjective & yes\\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &1\\ f(6)= &2\\ f(7)= &1\\ f(8)= &5\\ f(9)= &2\\ f(10)= &2\\ f(11)= &1\\ f(12)= &5\\ f(13)= &1\\ f(14)= &2\\ f(15)= &1\\ f(16)= &14\\ f(17)= &1\\ f(18)= &5\\ \end{array}$
\end{finite_members}
Information about small groups up to size 2000: http://www.tu-bs.de/~hubesche/small.html
\hyperbaseurl{http://math.chapman.edu/structures/files/}
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\begin{subclasses}\

\href{p-groups.pdf}{p-groups}

\href{nilpotent_groups.pdf}{nilpotent groups}

\href{solvable_groups.pdf}{solvable groups}

\end{subclasses}
\begin{superclasses}\

\href{Monoids.pdf}{Monoids}

\href{Inverse_semigroups.pdf}{Inverse semigroups}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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