Squares of characters and finite groups

Abstract

Let $G$ be a group of odd order and $\chi$ be a complex irreducible character. Then there exists a unique character $χ^{(2)}\in\Irr(G)$ such that $[{\chi }^2,{\chi }^{(2)}]$ is odd. Also, there exists a unique character $ψ\in \Irr(G)$ such that $[{\psi }^2,\chi ]$ is odd.

5 pages, corrected typos, added result