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Lattice Characterizations of 𝑝\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-Soluble and 𝑝\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p$$\end{document}-Supersoluble Finite Groups
被引：0
作者：
A-Ming Liu
[1
]
Sizhe Wang
[1
]
V. G. Safonov
[2
]
A. N. Skiba
[3
]
机构：
[1] School of Mathematics and Statistics,
[2] Hainan University,undefined
[3] Haikou,undefined
[4] School of Mathematics,undefined
[5] Tianjin University,undefined
[6] Institute of Mathematics,undefined
[7] National Academy of Sciences of Belarus,undefined
[8] Francisk Skorina Gomel State University,undefined
关键词：
finite group;
𝑝-soluble group;
𝑝-supersoluble group;
modular subgroup;
𝑁-modular subgroup;
D O I：
10.1134/S0081543824070125
中图分类号：
学科分类号：
摘要：
Let 𝐺\documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document} be a finite group, and let ℒ𝐺\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}}(G)$$\end{document} be the lattice of all subgroups of 𝐺\documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document}. A subgroup 𝑀\documentclass[12pt]{minimal}
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\begin{document}$$M$$\end{document} of 𝐺\documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document} is called modular in 𝐺\documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document} if 𝑀\documentclass[12pt]{minimal}
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\begin{document}$$M$$\end{document} is a modular element (in the Kurosh sense) of the lattice ℒ𝐺\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal{L}}(G)$$\end{document}, i.e., if (1) 𝑋𝑀𝑍𝑋𝑀𝑍\documentclass[12pt]{minimal}
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\begin{document}$$\langle X,M\cap Z\rangle=\langle X,M\rangle\cap Z$$\end{document} for all formulae-sequence𝑋𝐺𝑍𝐺\documentclass[12pt]{minimal}
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\begin{document}$$X\leq G,Z\leq G$$\end{document} such that 𝑋𝑍\documentclass[12pt]{minimal}
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\begin{document}$$X\leq Z$$\end{document}, and (2) 𝑀𝑌𝑍𝑀𝑌𝑍\documentclass[12pt]{minimal}
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\begin{document}$$\langle M,Y\cap Z\rangle=\langle M,Y\rangle\cap Z$$\end{document} for all formulae-sequence𝑌𝐺𝑍𝐺\documentclass[12pt]{minimal}
\usepackage{amsmath}
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\begin{document}$$Y\leq G,Z\leq G$$\end{document} such that 𝑀𝑍\documentclass[12pt]{minimal}
\usepackage{amsmath}
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\begin{document}$$M\leq Z$$\end{document}.
If 𝐴\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} is a subgroup of 𝐺\documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document}, then subscript𝐴𝑚𝐺\documentclass[12pt]{minimal}
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\begin{document}$$A_{mG}$$\end{document} is the subgroup of 𝐴\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} generated by all its subgroups that are modular in 𝐺\documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document}. We say that a subgroup 𝐴\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document} is 𝑁\documentclass[12pt]{minimal}
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\begin{document}$$N$$\end{document}-modular in 𝐺\documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document} (𝑁𝐺\documentclass[12pt]{minimal}
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\begin{document}$$N\leq G$$\end{document}) if, for some modular subgroup 𝑇\documentclass[12pt]{minimal}
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\begin{document}$$T$$\end{document} of 𝐺\documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document} containing 𝐴\documentclass[12pt]{minimal}
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\begin{document}$$A$$\end{document}, 𝑁\documentclass[12pt]{minimal}
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\begin{document}$$N$$\end{document}avoids the pair𝑇subscript𝐴𝑚𝐺\documentclass[12pt]{minimal}
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\begin{document}$$(T,A_{mG})$$\end{document}, i.e., 𝑁𝑇𝑁subscript𝐴𝑚𝐺\documentclass[12pt]{minimal}
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\begin{document}$$N\cap T=N\cap A_{mG}$$\end{document}. Using these notions, we give new characterizations of 𝑝\documentclass[12pt]{minimal}
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\begin{document}$$p$$\end{document}-soluble and 𝑝\documentclass[12pt]{minimal}
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\begin{document}$$p$$\end{document}-supersoluble finite groups.
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页码：S174 / S181
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