### 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 |
---|---|

Pages (from-to) | 203-218 |

Number of pages | 16 |

Journal | Journal of Algebra |

Volume | 388 |

DOIs | |

Publication status | Published - 2013 Aug 15 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

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

**Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3.** / Liu, Chia Hsin; Passman, D. S.

Research output: Contribution to journal › Article

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

}

TY - JOUR

T1 - Multiplicative Jordan decomposition in group rings with a Wedderburn component of degree 3

AU - Liu, Chia Hsin

AU - Passman, D. S.

PY - 2013/8/15

Y1 - 2013/8/15

N2 - 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.

AB - 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.

KW - 3-Group

KW - Central unit

KW - Integral group ring

KW - Multiplicative Jordan decomposition

KW - Wedderburn component

KW - Z-group

UR - http://www.scopus.com/inward/record.url?scp=84892504528&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892504528&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2013.04.015

DO - 10.1016/j.jalgebra.2013.04.015

M3 - Article

AN - SCOPUS:84892504528

VL - 388

SP - 203

EP - 218

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -