# 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 $G$.