Permutable complements

This article defines a symmetric relation on the collection of subgroups inside the same group.

Definition

Symbol-free definition

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 $H$ and $K$ of a group $G$ are termed permutable complements if the following two conditions hold:

• $H\cap K$ is the trivial group
• $HK=G$

Facts

Permutable complements need not be unique

Given a subgroup $H$ of $G$, there may or may not exist permutable complements of $H$. Moreover, there may exist multiple possibilities for a complement to $H$, and the multiple possibilities might not be pairwise isomorphic.

Further information: Every group of given order is a permutable complement for symmetric groups, Retract not implies normal complements are isomorphic

For a normal subgroup, they are fixed upto isomorphism

Interestingly, when a subgroup is normal, then any two permutable complements to it must be isomorphic. In fact, any permutable complement to it must be isomorphic to the quotient group.

Other related facts

• 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