Permutable complements

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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