Characters induced from Sylow subgroups (Q1270054)

Let \(G\) be a finite group and let \(p\) be a prime dividing \(| G|\). The paper deals with the question: What can be said about the structure of \(G\) if there exists a \(\chi\in\text{Irr}(G)\) which is induced from a Sylow-\(p\)-subgroup of \(G\) or equivalently, for which \(| G|/\chi(1)\) is a power of \(p\). (\(\chi\) is called \(p\)-induced.) In Theorem B, the authors prove that the condition above implies that \(G\) is \(p\)-constrained with \(O_{p'}(G)=1\). The crucial point here is to get \(p\)-constrainedness from the facts that \(G\) has only one \(p\)-block and \(O_{p'}(G)=1\), two conditions which are proved under the assumption of the existence of a \(p\)-induced irreducible character. At this point the classification of finite simple groups comes in and for \(p\) odd, \(p\)-constrainedness follows easily since non-Abelian simple groups have more than one \(p\)-block apart from the groups \(M_{22}\), \(M_{24}\) for \(p=2\) [already proved in the reviewer's paper J. Algebra 104, 135-140 (1986; Zbl 0618.20007)]. The characteristic two case is ruled out since \(M_{22}\) and \(M_{24}\) don't have 2-induced irreducible characters. Now Theorem B implies immediately Theorem A: If \(G\) has a \(p\)-induced irreducible character, then the center of a Sylow \(p\)-subgroup of \(G\) is subnormal in \(G\). The second part of the paper is devoted to the existence of \(p\)-induced irreducible characters. The authors prove Theorem C: Given any finite group \(X\) there exists a finite \(\mathbb{F}_pX\)-module \(V\) such that for any group extension \(V\rightarrowtail G\twoheadrightarrow X\) the group \(G\) admits a \(p\)-induced irreducible character. This means that the existence of a \(p\)-induced irreducible character does not restrict the factor group \(G/O_p(G)\). The module \(V\) is choosen as the permutation module over \(\mathbb{F}_p\) which is determined by the action of \(X\) on the right cosets of \(U\in\text{Syl}_p(X)\). For groups \(X\) of adjoint Lie-type of characteristic \(p\) the Steinberg module in characteristic \(p\) also suffices. Reviewer's remarks: a) Theorem B may be seen as the character theoretical analogue of Burnside's theorem that a group can not be non-Abelian simple if it has a conjugacy class length which is a nontrivial power of a prime \(p\). b) The classification of finite non-Abelian simple groups which have exactly one \(p\)-block (given in the Appendix) is already contained in \textit{M. E. Harris}' paper in J. Algebra 94, 411-424 (1985; Zbl 0602.20013).