Double coset index two implies maximal

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subgroup of double coset index two) must also satisfy the second subgroup property (i.e., maximal subgroup)
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Statement

If the Double coset index (?) of a subgroup in a group is two, i.e., if the subgroup has precisely two double cosets, then the subgroup is a Maximal subgroup (?).

Related facts

Proof

Proof idea

The idea is that any intermediate subgroup containing the given subgroup must be a union of double cosets of the subgroup.