Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3

Chia Hsin Liu, D. S. Passman

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

If G is a finite group whose integral group ring Z[G] has the multiplicative Jordan decomposition property, then it is known that all Wedderburn components of the rational group ring Q[G] have degree at most 3. While degree 3 components can occur, we prove here that if they do, then certain central units in Z[G] cannot exist. With this, we are able to greatly simplify the argument that characterizes those 3-groups with integral group ring having MJD. Furthermore, we show that if G is a nonabelian semidirect product of the form Cp⋊C3k, with prime p > 7 and with the cyclic 3-group acting like a group of order 3, then Z[G] does not have MJD.

Original languageEnglish
Pages (from-to)203-218
Number of pages16
JournalJournal of Algebra
Volume388
DOIs
Publication statusPublished - 2013 Aug 15

Keywords

  • 3-Group
  • Central unit
  • Integral group ring
  • Multiplicative Jordan decomposition
  • Wedderburn component
  • Z-group

ASJC Scopus subject areas

  • Algebra and Number Theory

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