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The Jordan-Hölder theorem with uniqueness for groups and semimodular lattices
被引：0
|作者：
Gábor Czédli
E. Tamás Schmidt
机构：
[1] University of Szeged,
[2] Bolyai Institute,undefined
[3] Mathematical Institute of the Budapest University of Technology and Economics,undefined
来源：
Algebra universalis
|
2011年
/
66卷
关键词：
Primary: 06C10;
Secondary: 20D35;
Jordan-Hölder theorem;
semimodular lattice;
slim lattice;
planar lattice;
group;
composition series;
subnormal subgroup;
chief series;
D O I：
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中图分类号：
学科分类号：
摘要：
For subnormal subgroups \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${A{\vartriangleleft}B}$$\end{document} and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${C{\vartriangleleft}D}$$\end{document} of a given group G, the factor B/A will be called subnormally down-and-up projective to D/C if there are subnormal subgroups \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${X{\vartriangleleft}Y}$$\end{document} such that \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${AY = B, A \cap Y = X, CY = D}$$\end{document} , and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${C \cap Y = X}$$\end{document} . Clearly, \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${B/A \cong D/C}$$\end{document} in this case. As G. Grätzer and J. B. Nation have recently pointed out, the standard proof of the classical Jordan-Hölder theorem yields somewhat more than is widely known; namely, the factors of any two given composition series are the same up to subnormal down-and-up projectivity and a permutation. We prove the uniqueness of this permutation.
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