Subgroup having a symmetric 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

A subgroup H of a group G is termed a subgroup having a symmetric transversal if there exists a symmetric subset S of G such that S is a left transversal of G. Note that for a symmetric subset, being a left transversal is equivalent to being a right transversal, because left and right coset spaces are naturally isomorphic via the bijection induced by the inverse map.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normal subgroup the left cosets are the same as the right cosets every left transversal is a right transversal. Note that existence of transversals requires the axiom of choice |FULL LIST, MORE INFO
Permutably complemented subgroup has a permutable complement we can take the permutable complement as the left transversal that is also a right transversal Subgroup having a 1-closed transversal, Subgroup having a twisted subgroup as transversal|FULL LIST, MORE INFO
Retract has a normal complement (via permutably complemented, via subset-conjugacy-closed) Subgroup having a 1-closed transversal, Subgroup having a twisted subgroup as transversal|FULL LIST, MORE INFO
Subgroup having a 1-closed transversal has a left transversal (equivalently, right transversal) that is a 1-closed subset, i.e., a union of subgroups |FULL LIST, MORE INFO
Subgroup having a transversal comprising involutions has a left transversal (equivalently, right transversal) all of whose non-identity elements are involutions Subgroup having a 1-closed transversal|FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Subgroup having a left transversal that is also a right transversal