Finite index in finite double coset index implies finite double coset index
This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Subgroup of finite index (?) and Subgroup of finite double coset index (?)), to another known subgroup property (i.e., Subgroup of finite double coset index (?))
View a complete list of composition computations
Statement with symbols
Suppose are groups such that has finite index in , and has finite Double coset index (?) in : there are double cosets of in . Then, has finite double coset index in , and the double coset index of in is bounded by:
Other facts about index and double coset index
- Index is multiplicative: If , then in the sense of possibly infinite cardinals. In particular, if has finite index in and has finite index in , then has finite index in .
- Finite double coset index is not transitive: We can have a situation of groups such that has finite double coset index in and has finite double coset index in , but does not have finite double coset index in .
Tightness of the result
For equal to the dihedral group of order for an odd prime , of order two, and trivial, the bound is tight. More generally, the bound is tight when is a Frobenius group, is a Frobenius complement in , and is the trivial subgroup.