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

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

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

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

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

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

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

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

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.