Permutable complements: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
Two | Two subgroups of a group are said to be permutable complements if: | ||
* Their intersection is trivial | * Their intersection is trivial | ||
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===Permutable complements need not be unique=== | ===Permutable complements need not be unique=== | ||
Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities | Given a subgroup <math>H</math> of <math>G</math>, there may or may not exist permutable complements of <math>H</math>. Moreover, there may exist multiple possibilities for a complement to <math>H</math>, and the multiple possibilities might not be pairwise isomorphic. | ||
{{further|[[Every group of given order is a permutable complement for symmetric groups]]}} | {{further|[[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=== | ===For a normal subgroup, they are fixed upto isomorphism=== | ||
Interestingly, when a subgroup is [[normal subgroup|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]]. | Interestingly, when a subgroup is [[normal subgroup|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]] | |||
Latest revision as of 20:49, 22 October 2016
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 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