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\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 |
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\Large Nilpotent groups \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Nilpotent_groups}{edit} |
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\abbreviation{Abbr} |
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\abbreviation{NGrp} |
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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} |
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A nilpotent group is a \href{Groups.pdf}{group} $\mathbf{G}=\langle G,\cdot,^{-1},1\rangle$ that is |
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$op_1$ is (name of property): $axiom_1$ |
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nilpotent: if $Z_0=\{1\}$ and $\forall i(Z_{i+1}=\{x \in G : \forall y\ xyx^{-1}y^{-1} \in Z_i\})$ then $\exists n(Z_n=G)$ |
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$op_2$ is ...: $...$ |
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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. |
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Remark: This is a template. |
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This is a template. |
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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: $h(x ... y)=h(x) ... h(y)$ |
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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: $h(x \cdot y)=h(x) \cdot h(y)$ |
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Classtype & (value, see description) \cite{Ln19xx} \\\hline |
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Classtype & higher-order \\\hline |
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Congruence modular & \\\hline Congruence $n$-permutable & \\\hline Congruence regular & \\\hline Congruence uniform & \\\hline |
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Congruence modular & yes \\\hline Congruence $n$-permutable & yes, $n=2$\\\hline Congruence regular & yes \\\hline Congruence uniform & yes \\\hline |
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\href{....pdf}{...} subvariety \href{....pdf}{...} expansion |
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\href{Abelian_groups.pdf}{Abelian groups} |
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\href{....pdf}{...} supervariety \href{....pdf}{...} subreduct |
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\href{Solvable_groups.pdf}{Solvable groups} supervariety |
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\bibitem{Ln19xx} |
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\bibitem{Lastname19xx} |
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