So it is a well know result that the quotient space $G/\cal R$ corresponding to right congruent relation $\cal R$ is equipotent to the quotient space $G/\cal L$ corresponding to the left congruent relation $\cal L$ so that is it is usual to use the simbol $|G:H|$ to indicate the cardinal correspondint to the two above quotient spaces.
So let's we prove the following theorem.
Theorem -Lagrange's Theorem
If $(H,\star)$ is a subgroup of a finite group $(G,\star)$ then the equality $$ |G|=|H|\cdot|G:H| $$ holds.
Proof. So we know that the quotient space of any equivalence relation is a partition so that the equality $$ |G|=\sum_{X\in G/\mathcal L}|X| $$ holds but for any $X\in G/\mathcal L$ there exists $x\in G$ such that $$ X=xH $$ so that actually for any $X\in G/\mathcal L$ the equality $$ |X|=|H| $$ holds and thus finally we conclude that $$ |G|=\sum_{X\in G/\mathcal L}|X|=\sum_{X\in G/\mathcal L}|H|=|H|\cdot|G/\mathcal L|=|H|\cdot|G:H| $$ as we desired.
So I would know why the last theorem is only true for finite groups: indeed, by cardinal arithmetic we know that the cardinality of any disjoint union is just (by definition, right?) the sum of cardinality of addends and just by the cardinal arithmetic we know that if $\lambda$ and $\kappa$ are cardinal the equality $$ \sum_{\alpha\in\lambda}\kappa=\kappa\cdot\lambda $$ holds. Moreover, I point out in this answer it seems implicitly that the above theorem is still valid for infinite group but unfortunately I did not find any textbook which say this so that I thought to put a specific question. So could someone help me, please?