Mathematical Structures: Nilpotent groups

# Nilpotent groups

Difference (from prior major revision) (no other diffs)

Changed: 28,30c28,29
 \Large Name of class \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Template}{edit} % Note: replace "Template" with Name_of_class in previous line

Changed: 32c31
 \abbreviation{Abbr}
 \abbreviation{NGrp}

Changed: 35,38c34
 A ... is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that $\langle A,...\rangle$ is a \href{Name_of_class.pdf}{name of class}
 A nilpotent group is a \href{Groups.pdf}{group} $\mathbf{G}=\langle G,\cdot,^{-1},1\rangle$ that is

Changed: 40c36
 $op_1$ is (name of property): $axiom_1$
 nilpotent: if $Z_0=\{1\}$ and $\forall i(Z_{i+1}=\{x \in G : \forall y\ xyx^{-1}y^{-1} \in Z_i\})$ then $\exists n(Z_n=G)$

Changed: 42c38,39
 $op_2$ is ...: $...$
 Remark: Note that $Z_1=Z(G)$, the center of $G$. The smallest $n$ for which $Z_n=G$ is the nilpotence class of $G$. E.g. Abelian groups are of nilpotence class 1.

Changed: 44c41
 Remark: This is a template.
 This is a template.

Changed: 51,52c48,49
 Let $\mathbf{A}$ and $\mathbf{B}$ be ... . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x ... y)=h(x) ... h(y)$
 Let $\mathbf{A}$ and $\mathbf{B}$ be nilpotent groups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \cdot y)=h(x) \cdot h(y)$

Changed: 78c75
 Classtype & (value, see description) \cite{Ln19xx} \\\hline
 Classtype & higher-order \\\hline

Changed: 85,88c82,85
 Congruence modular & \\\hline Congruence $n$-permutable & \\\hline Congruence regular & \\\hline Congruence uniform & \\\hline
 Congruence modular & yes \\\hline Congruence $n$-permutable & yes, $n=2$\\\hline Congruence regular & yes \\\hline Congruence uniform & yes \\\hline

Changed: 120,122c117
 \href{....pdf}{...} subvariety \href{....pdf}{...} expansion
 \href{Abelian_groups.pdf}{Abelian groups}

Changed: 128,130c123
 \href{....pdf}{...} supervariety \href{....pdf}{...} subreduct
 \href{Solvable_groups.pdf}{Solvable groups} supervariety

Changed: 136c129
 \bibitem{Ln19xx}
 \bibitem{Lastname19xx}

http://mathcs.chapman.edu/structuresold/files/Nilpotent_groups.pdf
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\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
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\begin{document}
\textbf{\Large Nilpotent groups}

\abbreviation{NGrp}

\begin{definition}
A \emph{nilpotent group} is a \href{Groups.pdf}{group} $\mathbf{G}=\langle G,\cdot,^{-1},1\rangle$ that is

\emph{nilpotent}:  if $Z_0=\{1\}$ and $\forall i(Z_{i+1}=\{x \in G : \forall y\ xyx^{-1}y^{-1} \in Z_i\})$ then $\exists n(Z_n=G)$

Remark: Note that $Z_1=Z(G)$, the center of $G$. The smallest $n$ for which $Z_n=G$ is the \emph{nilpotence class of $G$}. E.g. Abelian
groups are of nilpotence class 1.

This is a template.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be nilpotent groups. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$h(x \cdot y)=h(x) \cdot h(y)$
\end{morphisms}

\begin{definition}
An \emph{...} is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is ...:  $axiom$

$...$ is ...:  $axiom$
\end{definition}

\begin{basic_results}
\end{basic_results}

\begin{examples}
\begin{example}
\end{example}
\end{examples}

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

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
Classtype                       & higher-order \\\hline
Equational theory               & \\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & \\\hline
Residual size                   & \\\hline
Congruence distributive         & \\\hline
Congruence modular              & yes \\\hline
Congruence $n$-permutable       & yes, $n=2$\\\hline
Congruence regular              & yes \\\hline
Congruence uniform              & yes \\\hline
Congruence extension property   & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong.     & \\\hline
Amalgamation property           & \\\hline
Strong amalgamation property    & \\\hline
Epimorphisms are surjective     & \\\hline
\end{tabular}
\end{properties}
\end{table}

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

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

\end{finite_members}

\begin{subclasses}\

\href{Abelian_groups.pdf}{Abelian groups}

\end{subclasses}

\begin{superclasses}\

\href{Solvable_groups.pdf}{Solvable groups} supervariety

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Lastname19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview}

\end{thebibliography}

\end{document}
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