Permutable complements
This article defines a symmetric relation on the collection of subgroups inside the same group.
Contents
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 and
of a group
are termed permutable complements if the following two conditions hold:
-
is the trivial group
-
Facts
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.
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.
- 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