# Difference between revisions of "Subgroup of double coset index two"

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==Relation with other properties== | ==Relation with other properties== | ||

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+ | ===Stronger properties=== | ||

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+ | * [[Weaker than::Subgroup of index two]] | ||

+ | * Non-normal subgroup of index three: {{proofat|Index three implies normal or double coset index two]]}} | ||

+ | * Subgroup of index four that is not [[2-subnormal subgroup|2-subnormal]]: {{proofat|[[Index four implies 2-subnormal or double coset index two]]}} | ||

===Weaker properties=== | ===Weaker properties=== | ||

− | * [[Double coset-separated subgroup]] | + | * [[Stronger than::Double coset-separated subgroup]] |

− | * [[Double coset-ordering subgroup]] | + | * [[Stronger than::Double coset-ordering subgroup]] |

+ | * [[Stronger than::Maximal subgroup]]: {{proofofstrictimplicationat|[[Double coset index two implies maximal]]|[[Maximal not implies double coset index two]]}} | ||

+ | * [[Stronger than::1-completed subgroup]] | ||

+ | * [[Stronger than::Subgroup of finite double coset index]] | ||

+ | * [[Stronger than::Elliptic subgroup]] |

## Revision as of 15:47, 31 October 2008

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

A subgroup is said to have **double coset index two** if its double coset index is exactly two.

## Relation with other properties

### Stronger properties

- Subgroup of index two
- Non-normal subgroup of index three:
`For full proof, refer: Index three implies normal or double coset index two]]` - Subgroup of index four that is not 2-subnormal:
`For full proof, refer: Index four implies 2-subnormal or double coset index two`

### Weaker properties

- Double coset-separated subgroup
- Double coset-ordering subgroup
- Maximal subgroup:
*For proof of the implication, refer Double coset index two implies maximal and for proof of its strictness (i.e. the reverse implication being false) refer Maximal not implies double coset index two*. - 1-completed subgroup
- Subgroup of finite double coset index
- Elliptic subgroup