Subgroup having a 1-closed transversal
From Groupprops
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]
Contents
Definition
No. | Shorthand | A subgroup of a group is termed a subgroup having a 1-closed transversal if ... | A subgroup of a group is termed a subgroup having a 1-closed transversal if ... |
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1 | left transversal that is 1-closed | it has a left transversal that is a 1-closed subset, i.e., is a union of subgroups of | there exists a subset such that intersects every left coset of in exactly one element, and is a union of subgroups of |
2 | right transversal that is 1-closed | it has a left transversal that is a 1-closed subset, i.e., is a union of subgroups of | there exists a subset such that intersects every right coset of in exactly one element, and is a union of subgroups of |
3 | left transversal + right transversal + 1-closed | it has a left transversal that is also a right transversal and is also 1-closed | there exists a subset such that intersects every left coset of in exactly one element, intersects every right coset of in exactly one element, and is a union of subgroups of |
Examples
- Any permutably complemented subgroup has a 1-closed transversal, because we can take the transversal as the complementary subgroup.
- In a group of prime exponent, any subgroup has a 1-closed transversal.
Relation with other properties
Stronger properties
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
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Subgroup having a left transversal that is also a right transversal | has a left transversal that is also a right transversal | Subgroup having a symmetric transversal|FULL LIST, MORE INFO | ||
Subgroup having a symmetric transversal | has a left transversal that is a symmetric subset | follows because any 1-closed subset is symmetric | |FULL LIST, MORE INFO |