Mathematical Structures: History of Nilpotent groups

# History of Nilpotent groups

 Revision 2 . . July 31, 2004 8:13 pm by Jipsen Revision 1 . . July 9, 2004 10:25 am by Jipsen

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}