This article defines a symmetric relation on the collection of subgroups inside the same group.
Two subgroups of a group are said to be permutable complements if:
- Their intersection is trivial
- Their product is the whole group
Definition with symbols
Two subgroups and of a group are termed permutable complements if the following two conditions hold:
- is the trivial group
Permutable complements need not be unique
Given a subgroup of , there may or may not exist permutable complements of . Moreover, there may exist multiple possibilities for a complement to , and the multiple possibilities might not be pairwise isomorphic.
For a normal subgroup, they are fixed upto isomorphism
- There may be multiple subgroups that are pairwise permutable complements
- Retract not implies every permutable complement is normal
- Permutable complement to normal subgroup is isomorphic to quotient
- Permutable complements to abelian normal subgroup are automorphic
- Retract not implies normal complements are isomorphic
- Permutable complements to normal subgroup need not be automorphic