Abstract
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.
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Presented by Radha Kessar.
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Lewis, M.L., Prajapati, S.K. On the Existence of Johnson Polynomials for Nilpotent Groups. Algebr Represent Theor 18, 205–213 (2015). https://doi.org/10.1007/s10468-014-9488-5
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DOI: https://doi.org/10.1007/s10468-014-9488-5