A character theoretic criterion for Fitting height

For a finite group G, setIrr N ( G ) := { chi is an element of Irr ( G ) divided by chi ( 1 ) 2 inverted iota | G : ker chi | } , cd N ( G ) := { chi ( 1 ) divided by chi is an element of Irr N ( G ) } .In this paper, we study finite groups G with | cd N ( G ) | <= 1. In particular, we prove that if G has odd order, then h ( G ) <= | cd N ( G ) | + 1, where h ( G ) h(G) is the Fitting height of G.