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Stepanov, Alexey. (1394). Sandwich classification theorem. , 4(3), 7-12. doi: 10.22108/ijgt.2015.10515
Alexey Stepanov. "Sandwich classification theorem". , 4, 3, 1394, 7-12. doi: 10.22108/ijgt.2015.10515
Stepanov, Alexey. (1394). 'Sandwich classification theorem', , 4(3), pp. 7-12. doi: 10.22108/ijgt.2015.10515
Stepanov, Alexey. Sandwich classification theorem. , 1394; 4(3): 7-12. doi: 10.22108/ijgt.2015.10515

Sandwich classification theorem

International Journal of Group Theory
مقاله 3، دوره 4، شماره 3، آذر 2015، صفحه 7-12    اصل مقاله (188.83 K)
نوع مقاله: Ischia Group Theory 2014
شناسه دیجیتال (DOI): 10.22108/ijgt.2015.10515
نویسنده
Alexey Stepanov*
St.Petersburg State University
چکیده
‎The present note arises from the author's talk at the conference ``Ischia Group Theory 2014''‎. ‎For subgroups $F\le N$ of a group $G$ denote by $L(F,N)$‎ ‎the set of all subgroups of $N$‎, ‎containing $F$‎. ‎Let $D$ be a subgroup of $G$‎. ‎In this note we study the lattice $LL=L(D,G)$ and the lattice $LL'$‎ ‎of subgroups of $G$‎, ‎normalized by $D$‎. ‎We say that $LL$ satisfies sandwich classification‎ ‎theorem if $LL$ splits into a disjoint union of sandwiches $L(F,N_G(F))$‎ ‎over all subgroups $F$ such that the normal closure of $D$ in $F$ coincides with $F$‎. ‎Here $N_G(F)$ denotes the normalizer of $F$ in $G$‎. ‎A similar notion of sandwich classification‎ ‎is introduced for the lattice $LL'$‎. ‎If $D$ is perfect‎, ‎i.\,e‎. ‎coincides with its commutator‎ ‎subgroup‎, ‎then it turns out that sandwich classification theorem for $LL$ and $LL'$ are equivalent‎. ‎We also show how to find basic subroup $F$ of sandwiches for $LL'$ and review‎ ‎sandwich classification theorems in algebraic groups over rings‎.
کلیدواژه‌ها
subgroup structure؛ sandwich classification؛ Chevalley groups؛ commutative rings
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