Orders of units in integral group rings and blocks of defect $1$
Abstract
We show that if the Sylow $p$-subgroup of a finite group $G$ is of order $p$, then the normalized unit group of the integral group ring of $G$ contains a normalized unit of order $pq$ if and only if $G$ contains an element of order $pq$, where $q$ is any prime. We use this result to answer the Prime Graph Question for most sporadic simple groups and some simple groups of Lie type, including a new infinite series of such groups. Our methods are based on the understanding of blocks of cyclic defect and Young tableaux combinatorics.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2019
- DOI:
- 10.48550/arXiv.1906.03570
- arXiv:
- arXiv:1906.03570
- Bibcode:
- 2019arXiv190603570C
- Keywords:
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- Mathematics - Group Theory;
- 16U60;
- 20C05;
- 20C20;
- 05E10
- E-Print:
- 32 pages. Minor corrections and clarifications according to referee's comments