The 2-part of the permutation index of a group character (Q1076795)

A theorem due to E. Artin says the following. There exists a least positive number n(\(\chi)\) such that n(\(\chi)\)sp(\(\chi)\) is a difference of permutation characters of G. Here sp(\(\chi)\) is the sum of the distinct Galois conjugates of the irreducible complex character \(\chi\) of G. This paper gives the construction of the following fact. For every q, a positive power of 2, there is a solvable group G and \(\chi\in Irr G\) such that \(\chi\) (1) is odd, yet n(\(\chi)\) is divisible by q. \{In the author's paper [ibid. 93, 445-474 (1985; Zbl 0596.20006)] it is proved that, for each solvable group H, the odd part of \(n(\eta)\) divides \(\eta\) (1) whenever \(\eta\in Irr H\).\}