Homogeneous character induction
Let \(G\) be a finite group and let \(H\) be a proper subgroup of \(G\). The condition (CI) is said to hold if each non-principal irreducible character \(\theta\) of \(H\) induces homogeneously to \(G\); i.e., \(\theta^ G\) is an integral multiple of some irreducible character of \(G\). The authors first show that if \(H\) is a normal subgroup of \(G\), then (CI) holds if and only if \((G,H)\) is a Camina pair, which is to say, \(| C_ G(x)|=| C_{G/H}(xH)|\) for all \(x\notin H\). The bulk of the paper is devoted to the case where \(H\) is definitely not normal in \(G\) and (CI) holds. In this situation, \((G,\text{clos}_ G(H))\) is a Camina pair, where \(\text{clos}_ G(H)\) is the normal closure of \(H\) in \(G\). Further, \(H\) is nilpotent, and either \(G\) is a Frobenius group with Frobenius kernel \(\text{clos}_ G(H)\) or \(\text{clos}_ G(H)\) is a \(p\)-group. In the first of these cases, \(G\) is doubly transitive. It follows that if (CI) holds, then \(H\) is subnormal in \(G\), and if not normal, it must be a \(p\)-group. Some special information is also given in the case where \(N=\text{clos}_ G(H)\) is a \(p\)-group and \(p\) is a divisor of \(| G/N|\).
scientific article; zbMATH DE number 63622