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Ultrasoluble coverings of some nilpotent groups by a cyclic group
over number fields and related questions
D. D. Kiselev All-Russian Academy of International Trade
Abstract:
-Let $F$ be a finite nilpotent group of odd order. For every finite cyclic
subgroup $A$ of odd order we find necessary and sufficient conditions
for a class $h\in H^2(F,A)$ to determine an ultrasoluble extension (under the
additional assumption of minimality of all $p$-Sylow subextensions to
the extension with class $h$ for all non-Abelian $p$-Sylow subgroups
$F_p$ of $F$), that is, for the existence of a Galois extension of number fields
$K/k$ with group $F$ such that the corresponding embedding problem is
ultrasoluble (it has solutions and all such solutions are fields). We also
establish a number of related results.
Keywords:
-embedding problem, concordance condition, ultrasolubility, co-embedding problem.
DOI:
https://doi.org/10.4213/im8636
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English version:
Izvestiya: Mathematics, 2018, 82:3, 512–531
Bibliographic databases:
UDC:
512.623.32
MSC: 12F12, 11R32, 16K50 Received: 05.12.2016 Revised: 09.04.2017
Citation:
D. D. Kiselev, “Ultrasoluble coverings of some nilpotent groups by a cyclic group
over number fields and related questions”, Izv. RAN. Ser. Mat., 82:3 (2018), 69–89; Izv. Math., 82:3 (2018), 512–531
Citation in format AMSBIB
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Linking options:
http://mi.mathnet.ru/eng/izv8636https://doi.org/10.4213/im8636 http://mi.mathnet.ru/eng/izv/v82/i3/p69
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