Some necessary conditions for compatibility of groups

Abstract Two groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> L_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> L_{2} are compatible if there exists a finite group 𝐺 with isomorphic normal subgroups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>N</m:mi> <m:mn>1</m:mn> </m:msub> </m:math> N_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>N</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> N_{2} such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>≅</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:msub> <m:mi>N</m:mi> <m:mn>1</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> L_{1}\cong G/N_{1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>L</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>≅</m:mo> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:msub> <m:mi>N</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:mrow> </m:math> L_{2}\cong G/N_{2} . In this paper, we give new necessary conditions for two groups to be compatible.