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
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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

• Subnormality is permuting upper join-closed in finite
• Subnormality is not upper join-closed
• 2-subnormality is not upper join-closed

References

Journal references

• Paper:CasoloDardano04^{More info}