Bound on double coset index in terms of orders of group and subgroup

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Statement

For general groups

Let G be a group, and H be a subgroup. Then, if d denotes the cardinality of the Double coset space (?) of H in G, i.e., the Double coset index (?) of H in G, we have:

1 + \frac{[G:H] - 1}{|H|} \le d \le [G:H].

When H is a Subgroup of finite index (?) in G, the left inequality attains equality if and only if H is a Malnormal subgroup (?), and the right inequality attains equality if and only if H is a Normal subgroup (?) of G.

For finite groups

Let G be a finite group, H be a subgroup. Then, if d denotes the cardinality of the Double coset space (?) of H in G, i.e., the Double coset index (?) of H in G, we have:

1 + \frac{|G|}{|H|^2} - \frac{1}{|H|} \le d \le \frac{|G|}{|H|}

The left inequality attains equality if and only if H is a Malnormal subgroup (?) (i.e., it is either equal to the whole group, or is a Frobenius subgroup (?) in the group), and the right inequality attains equality if and only if H is a Normal subgroup (?) of G.