Conjugation-invariantly permutably complemented subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]


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Definition

Symbol-free definition

A subgroup of a group is said to be conjugation-invariantly permutably complemented if it satisfies both the conditions below:

  • It is permutably complemented, viz the set of its permutable complements is empty
  • The set of its permutable complements is closed under conjugation. In other words, any conjugate of a permutable complement is also a permutable complement.

Definition with symbols

A subgroup H of a group G is said to be conjugation-invariantly permutably complemented if the following are true:

  • There exists a subgroup K of G such that HK = G and H \cap K is trivial (in other words, a permutable complement of H in G)
  • If K and H are permutable complements, then K^g and H are also permutable complements for any g \in G.

Relation with other properties

Stronger properties

Weaker properties