### 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 language | English |
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Pages (from-to) | 203-218 |

Number of pages | 16 |

Journal | Journal of Algebra |

Volume | 388 |

DOIs | |

Publication status | Published - 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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## Cite this

Liu, C. H., & Passman, D. S. (2013). Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3.

*Journal of Algebra*,*388*, 203-218. https://doi.org/10.1016/j.jalgebra.2013.04.015