An application of blocks to torsion units in group rings

Bächle, Andreas; Margolis, Leo

doi:10.1090/proc/14561

An application of blocks to torsion units in group rings
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by Andreas Bächle and Leo Margolis PDF

Proc. Amer. Math. Soc. 147 (2019), 4221-4231 Request permission

Abstract:

We use the theory of blocks of cyclic defect to prove that under a certain condition on the principal $p$-block of a finite group $G$, the normalized unit group of the integral group ring of $G$ contains an element of order $pq$ if and only if so does $G$ for $q$ a prime different from $p$. Using this we verify the Prime Graph Question for all alternating and symmetric groups and also for two sporadic simple groups.

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