Finite groups whose subgroup graph contains a vertex of large degree

Abstract Burness and Scott (J Aust Math Soc 87:329-357, 2009) classified finite groups G such that the number of prime order subgroups of G is greater than $$|G|/2-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>G</mml:mi> <mml:mo>|</mml:mo> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . In this note, we study finite groups G whose subgroup graph contains a vertex of degree greater than $$|G|/2-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>G</mml:mi> <mml:mo>|</mml:mo> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . The classification given for finite solvable groups extends the work of Burness and Scott.