Subnormality is not permuting upper join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., subnormal subgroup) not satisfying a subgroup metaproperty (i.e., permuting upper join-closed subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about subnormal subgroup|Get more facts about permuting upper join-closed subgroup property|

This article gives the statement, and possibly proof, of a subgroup property (i.e., 2-subnormal subgroup) not satisfying a subgroup metaproperty (i.e., permuting upper join-closed subgroup property).
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about 2-subnormal subgroup|Get more facts about permuting upper join-closed subgroup property|

Statement

It is possible to have a group $G$ , a subgroup $H$ of $G$ , and intermediate subgroups $K_{1}$ and $K_{2}$ of $G$ containing $H$ such that $H$ is a 2-subnormal subgroup in both $K_{1}$ and $K_{2}$ , $K_{1} K_{2} = K_{2} K_{1}$ (i.e., they are permuting subgroups), but $H$ is not a subnormal subgroup of $K_{1} K_{2}$ .

Related facts

References

Journal references

Paper:CasoloDardano04^{More info}