On the Existence of Johnson Polynomials for Nilpotent Groups.

Let G be a finite group. We say that G has a Johnson polynomial if there exists a polynomial f( x) ∈ ℤ[ x] and a character χ ∈ Irr( G) so that f( χ) equals the total character for G. In this paper, we show that if G has nilpotence class 2, then G has a Johnson polynomial if and only if G is an extra-special 2-group. Generalizing this, we say that G has a generalized Johnson polynomial if f( x) ∈ ℚ[ x]. We show that if G has nilpotence class 2, then G has a generalized Johnson polynomial if and only if Z( G) is cyclic. Also, if G is nilpotent and |cd( G)| = 2, then G has a generalized Johnson polynomial if and only if G has nilpotence class 2 and Z( G) is cyclic. [ABSTRACT FROM AUTHOR]