### Abstract

The usual approach to group representations in modern texts is via the theory of algebras and modules. This chapter is based on a less well-known constructive approach to the theory of algebras which uses the generalization to noncommutative algebras of a (multiplicative) norm, which can be applied to obtain results on group determinants. This continues a line of research which goes back to Frobenius. Significant results which have not been translated into modern abstract accounts are to be found there. An essential result is that given a norm-type form on an algebra with suitable assumptions, the form can be constructed rationally from the first three coefficients of its “characteristic equation”. This, applied to the group determinant, shows that the 1-, 2- and 3-characters determine a group.

Original language | English (US) |
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Title of host publication | Lecture Notes in Mathematics |

Publisher | Springer Verlag |

Pages | 111-136 |

Number of pages | 26 |

DOIs | |

Publication status | Published - Jan 1 2019 |

### Publication series

Name | Lecture Notes in Mathematics |
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Volume | 2233 |

ISSN (Print) | 0075-8434 |

ISSN (Electronic) | 1617-9692 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Lecture Notes in Mathematics*(pp. 111-136). (Lecture Notes in Mathematics; Vol. 2233). Springer Verlag. https://doi.org/10.1007/978-3-030-28300-1_3