Groups whose real irreducible characters have degrees coprime to \(p\). (Q713398)

This paper is concerned with variations of the well-known result of Itô and Michler that a finite group \(G\) has an Abelian normal Sylow \(p\)-subgroup if and only if \(p\) does not divide any irreducible (complex) character degree of \(G\). A recent result of Tiep (no reference is given in the paper) states that if every real-valued irreducible character of the finite group \(G\) has \(p'\)-degree for an odd prime \(p\), then \(O^{p'}(G)\) is solvable. In the paper under review, the authors study what else can be said about the groups satisfying the hypothesis of Tiep's result, and they prove a number of results on this. They show, for example, that if we put \(K=O^{2'}(G)\), then \(K\) has a normal Sylow \(p\)-subgroup \(Q\), and \(Q'\) is central in \(K\). Also, if \(G/K\) has a normal Sylow \(p\)-subgroup, then \(G\) has a normal Sylow \(p\)-subgroup, and if \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(N_G(P)\) contains an element in each real conjugacy class of \(G\). Furthermore, a canonically defined bijection between the set of real-valued irreducible characters of \(G\) and the set of real-valued irreducible characters of \(p'\)-degree of \(N_G(P)\) is established.