##  [Index of a Subgroup](/index-subgroup-0) 

 Definition

The cardinal number of the set of left cosets (equivalently right cosets) of a subgroup H in a group G; it measures the relative size of H inside G and is finite when G decomposes into finitely many cosets of H.

 

 

 

 

 

 





## Principle

Principle

Index counts distinct cosets partitioning the group and satisfies multiplicative relations in towers: if K ≤ H ≤ G and indices are finite then [G:K] = [G:H]·[H:K].

 

 

 

 

 





## Demonstration

Demonstration

In Z the subgroup 2Z has index 2 because the cosets are 2Z and 1+2Z; in S4 the stabilizer of a point has index 4 since there are four images for that point under permutation action, giving four distinct cosets.

 

 

 

 

## Misapplication

Misapplication

Assuming index finiteness implies normality, or confusing index with subgroup order: a subgroup of index 2 is always normal but index 3 need not be; moreover infinite groups may have finite-index subgroups, so index finiteness does not imply finiteness of the subgroup itself.

 

 

 

 

 





## Consequence

Consequence

Finite index has many structural consequences: group actions on coset spaces, existence of finite-index normal cores, and restrictions on possible homomorphisms; in geometric group theory finite-index subgroups inherit many large-scale properties of the ambient group.

 

 

 

 

## Reversal

Reversal

The reverse viewpoint considers the subgroup by how sparse it is: trivial subgroup has maximal index equal to |G| in finite groups, while the whole group has index 1; reversing emphasizes extremes from dense (index 1) to sparse (large or infinite index).

 

 

 

 

 





## Boundary

Boundary

Defined for subgroups of groups; excludes general subsets, submonoids, or relations where coset partitioning fails; index can be infinite and different conventions may arise for topological or measured groups where “index” must account for measure or topology.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension appears between index and order of quotient groups: when H is normal, [G:H] equals |G/H|, but without normality index is still defined while quotient group is not; novices may conflate these notions.

 

 

 

 

 





## Synthesis

Synthesis

Index of a subgroup quantifies how a subgroup partitions the group into cosets, obeys multiplicative relations in subgroup chains, and controls many algebraic and geometric inheritance properties via finite-index arguments.