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Integral group rings of solvable groups with trivial central units

Integral group rings of solvable groups with trivial central units AbstractThe integral group ring ℤ⁢G{\mathbb{Z}G}of a group G has only trivial central units if the only central units of ℤ⁢G{\mathbb{Z}G}are ±z{\pm z}for z in the center of G. We show that the order of a finite solvable group G with this property can only be divisible by the primes 2, 3, 5 and 7, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Integral group rings of solvable groups with trivial central units

Forum Mathematicum , Volume 30 (4): 11 – Jul 1, 2018

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References (16)

Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1435-5337
eISSN
1435-5337
DOI
10.1515/forum-2017-0021
Publisher site
See Article on Publisher Site

Abstract

AbstractThe integral group ring ℤ⁢G{\mathbb{Z}G}of a group G has only trivial central units if the only central units of ℤ⁢G{\mathbb{Z}G}are ±z{\pm z}for z in the center of G. We show that the order of a finite solvable group G with this property can only be divisible by the primes 2, 3, 5 and 7, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units.

Journal

Forum Mathematicumde Gruyter

Published: Jul 1, 2018

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