Orthogonal subnormal subgroups permute

Name
This result is sometimes termed Roseblade's permutability theorem, after Roseblade, who proved it in 1965.

History
The statement was conjectured by Philip Hall in the light of Wielandt's weaker versions of the result (Wielandt proved similar results, but restricting attention to a group of finite composition length). It was proved by Roseblade in 1965.

Statement
Suppose $$H$$ and $$K$$ are fact about::subnormal subgroups of $$G$$ such that $$H$$ and $$K$$ are fact about::orthogonal groups: the tensor product of their Abelianizations is zero. Then, $$H$$ and $$K$$ are permuting subgroups.

Converse

 * Non-orthogonal groups can be embedded as non-permuting subnormal subgroups in some group: If $$H$$ and $$K$$ are two groups that are not orthogonal to each other, there exists a group $$G$$ with subgroups <math_0 \cong H, K_0 \cong K, such that $$H_0, K_0$$ are both subnormal in $$G$$, but $$H_0$$ and $$K_0$$ are not permuting subgroups.

Corollaries

 * Perfect subnormal implies subnormal-permutable: Any perfect subnormal subgroup permutes with any subnormal subgroup.
 * Perfect subnormal implies join-transitively subnormal