Irreducible constituents of induced monomial characters (Q1086671)

The following two results are proved: Th. A. Let G be a p-solvable finite group with \(p>2\) and let \(N\trianglelefteq G\) with G/N supersolvable of odd order, \(\pi\) (\(| G/N|)\subseteq \pi (p(p-1))\). Suppose \(\theta\in Irr(N)\) is monomial and \(\theta\) (1) is a power of p. Then every irreducible constituent of \(\theta^ G\) is monomial. - Th. B. Let G be as in Th. A and suppose \(\chi\in Irr(G)\) is primitive and \(\chi\) (1) is a power of p. Let \(N\trianglelefteq G\) and G/N be as in Th. A. Then \(\chi_ N\) is irreducible and primitive. Both theorems are false if \(p=2\) or if \(p>2\) and \(| G/N|\) is even. Th. A is a converse to Dade's following theorem: Let G be p- solvable, \(p>2\), and suppose \(\chi\in Irr(G)\) is monomial and \(\chi\) (1) is a power of p. Let S be a subnormal subgroup of G. Then every irreducible constituent of \(\chi_ S\) is monomial.