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

## Statement

### For general groups

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

.

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

### For finite groups

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

The left inequality attains equality if and only if 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 is a Normal subgroup (?) of .