Complements to abelian normal subgroup are automorphic
- The restriction of to is the identity map on .
- induces an isomorphism from to .
In particular, and are automorphic subgroups.
(Note: There may exist automorphic subgroups to that are not permutable complements to ).
- Complement to normal subgroup is isomorphic to quotient
- Hall retract implies order-conjugate: This is the conjugacy part of the Schur-Zassenhaus theorem, and states that, for a normal Hall subgroup, any two complements are not just automorphic, they are also conjugate subgroups.
- Complements to normal subgroup need not be automorphic
Breakdown when the normality constraint is removed or shifted
- Every group of given order is a permutable complement for symmetric groups: A complete breakdown of the analogous statement when the subgroup is no longer assumed to be normal.
- Retract not implies every permutable complement is normal
- Retract not implies normal complements are isomorphic