Real strongly monolithic characters of finite groups

Abstract In this paper, we are interested in the real-valued strongly monolithic characters of finite groups. We give some criteria for solvability and normal p -complements of finite groups by their real-valued strongly monolithic characters. We also prove that a finite nonsolvable group has a real-valued strongly monolithic character of degree $$\geqslant 5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⩾</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:math> . Additionally, we give a classification for odd-square-free groups by using their real-valued strongly monolithic characters and the zeros of their irreducible characters of odd degree.