Abstract
Let \(K:=\mathbb {Q}(G)\) be the number field generated by the complex character values of a finite group G. Let \(\mathbb {Z}_K\) be the ring of integers of K. In this paper we investigate the suborder \(\mathbb {Z}[G]\) of \(\mathbb {Z}_K\) generated by the character values of G. We prove that every prime divisor of the order of the finite abelian group \(\mathbb {Z}_K/\mathbb {Z}[G]\) divides |G|. Moreover, if G is nilpotent, we show that the exponent of \(\mathbb {Z}_K/\mathbb {Z}[G]\) is a proper divisor of |G| unless \(G=1\). We conjecture that this holds for arbitrary finite groups G.
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Acknowledgements
The work on this paper started with a visit of the first author at the University of Jena in January 2019. He appreciates the hospitality received there. The authors also like to thank Thomas Breuer for making them aware of the CoReLG package [2] of GAP [11] which was used for computations with alternating groups. The first author is a postdoctoral researcher of the FWO (Research Foundation Flanders). The second author is supported by the German Research Foundation (SA 2864/1-1 and SA 2864/3-1).
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Communicated by John S. Wilson.
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Bächle, A., Sambale, B. Orders generated by character values. Monatsh Math 191, 665–678 (2020). https://doi.org/10.1007/s00605-019-01324-3
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DOI: https://doi.org/10.1007/s00605-019-01324-3