Index of a subgroup in the context of "Gromov's theorem on groups of polynomial growth"

In mathematics, specifically group theory, the index of a subgroup H in a group G is the number of left cosets of H in G, or equivalently, the number of right cosets of H in G.The index is denoted ${\displaystyle |G:H|}$ or ${\displaystyle [G:H]}$ or ${\displaystyle (G:H)}$.Because G is the disjoint union of the left cosets and because each left coset has the same size as H, the index is related to the orders of the two groups by the formula

(interpret the quantities as cardinal numbers if some of them are infinite).Thus the index ${\displaystyle |G:H|}$ measures the "relative sizes" of G and H.