A Note on Cancellation for Frobenius Group Rings with Binary Polyhedral Complements

Abstract

Let \(G=N\rtimes H\) be a finite Frobenius group with nontrivial abelian kernel \(N\) and binary polyhedral complement \(H\). We record a character-theoretic observation concerning the part of \(\mathbb Q[G]\) supported on the nontrivial characters of \(N\): every corresponding induced irreducible character has Frobenius–Schur indicator \(+1\). Hence this kernel-induced factor contributes no totally definite quaternion component, and the relevant Hamilton quaternion multiplicity is inherited from \(H\). As an application of Nicholson’s comparison theorem and Swan’s classification, we recover the criterion\[\mathbb Z[G]\text{ has stably free cancellation}\quad\Longleftrightarrow\quadH\in\{Q_8,Q_{12},Q_{16},Q_{20},2T,2O,2I\}.\]We also note that this conclusion is independent of the abelian kernel and of the fixed-point-free action, and that a stably free nonfree module over \(\mathbb Z[H]\) remains nonfree after extension of scalars to \(\mathbb Z[G]\). Keywords Integral group rings; stably free cancellation; Frobenius groups; binary polyhedral groups; Frobenius–Schur indicators.