Subgroup having a twisted subgroup as transversal

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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]

Definition

Suppose H is a subgroup of a group G. We say that H is a subgroup having a twisted subgroup as transversal if there exists a twisted subgroup S of G that is also a left transversal for H in G, i.e., it intersects every left coset of H in G at exactly one point. Note that S will automatically also be a right transversal for H in G.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
permutably complemented subgroup |FULL LIST, MORE INFO
direct factor (via permutably complemented) (via permutably complemented) |FULL LIST, MORE INFO
retract (via permutably complemented) (via permutably complemented) |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
subgroup having a 1-closed transversal |FULL LIST, MORE INFO
subgroup having a symmetric transversal |FULL LIST, MORE INFO
subgroup having a left transversal that is also a right transversal Subgroup having a 1-closed transversal|FULL LIST, MORE INFO