Groups whose irreducible representations have degree at most 2. (Q555970)

Let \(G\) be a finite group, and \(F\) an algebraically closed field of characteristic zero. It is a result due to \textit{S. A. Amitsur} [Ill. J. Math. 5, 198-205 (1961; Zbl 0100.25704)] that all irreducible \(F\)-representations of \(G\) are of degree 1 or 2 if and only if (i) \(G\) is Abelian, or (ii) \(G\) has an Abelian subgroup of index 2, or (iii) \(G/\zeta(G)\simeq C_2\times C_2\times C_2\), where \(\zeta(G)\) is the centre of \(G\) and \(C_2\) is the cylic group of order 2. Extending this result to positive characteristic, the author proves that if \(\text{char}(F)>2\), then all \(F\)-representations of \(G\) are of degree 1 or 2 if and only if \(G\) has a normal Sylow \(p\)-subgroup \(P\), and \(G/P\) has one of the above three properties. If \(F\) is an algebraically closed field of characteristic different from 2, the finite groups \(G\) for which every irreducible \(F\)-representation of \(G\) is of degree 1 or 2 and the degree 2 representations are self-contragradient are classified. This result is then used to characterize the finite groups \(G\) for which the skew elements in the group algebra \(F[G]\) commute.