File:Hierarchy of finite groups.svg
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[edit]DescriptionHierarchy of finite groups.svg |
English: Hierarchy of fundamental classes (including simple groups, solvable groups, nilpotent group, p-groups, abelian groups, elementary abelian p-groups, and cyclic groups) of finite groups are displayed via Hasse diagram. The gaps between these classes are shown by concrete examples on the edges. |
Date | |
Source | Own work |
Author | ARAKI Satoru |
\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{positioning}
\begin{document} \begin{tikzpicture}
\node[label=above:finite groups]
(grp) {\( \circ \)};
\node[label=left:simple groups]
(simp) [below left=of grp] {\( \circ \)};
\node[label=right:solvable groups]
(solv) [below right=of grp] {\( \circ \)};
\node[label=right:nilpotent groups]
(nilp) [below right=of solv] {\( \circ \)};
\node[label=right:\( p \)-groups]
(p-gp) [below right =of nilp] {\( \circ \)};
\node[label=left:abelian groups]
(ab) [below left =of nilp] {\( \circ \)};
\node[label=right:elementary abelian \( p \)-groups]
(elem) [below left =of p-gp] {\( \circ \)};
\node[label=left:cyclic groups]
(cyc) [below left =of ab] {\( \circ \)};
\node[label=right:the trivial group]
(triv) [below right =of cyc] {\( \circ \)};
\tikzset{every node/.style={fill=white, scale=.8}}
\path
(grp) edge node {\( A_5 \)} (solv)
(grp) edge node {\( S_5 \)} (simp)
(solv) edge node {\( S_3 \)} (nilp)
(nilp) edge node {\( C_3 \times Q_8 \)} (p-gp)
(nilp) edge node {\( Q_8 \)} (ab)
(ab) edge node {\( C_3 \times V_4 \)} (cyc)
(ab) edge node {\( C_4 \)} (elem)
(p-gp) edge node {\( Q_8 \)} (elem)
(elem) edge node {\( V_4 \)} (triv)
(cyc) edge node {\( C_6 \)} (triv)
;
\end{tikzpicture} \end{document}
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[edit]I, the copyright holder of this work, hereby publish it under the following license:
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Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 14:04, 18 June 2017 | 412 × 285 (51 KB) | ARAKI Satoru (talk | contribs) | corrected errors | |
13:47, 18 June 2017 | 412 × 285 (52 KB) | ARAKI Satoru (talk | contribs) | User created page with UploadWizard |
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