History of Lagrange's theorem

Lagrange's theorem is currently stated as:

If $G$ is a finite group and $H$ is a subgroup of $G$, then:

$\left|G\right|=\left|H\right|[G:H]$

where $\left|G\right|,\left|H\right|$ are the orders of the groups and $[G:H]$ is the index of $H$ in $G$.

Lagrange's original motivation

Lagrange wanted to determine whether the quintic equation could be solved by means of radicals. He sought to reduce the problem of solving the quintic to solving an auxiliary equation of lower degree. For this, he considered rational functions of the roots and the set of values these rational functions could take for different permutations of the roots. He proved that the number of possible values these rational functions can take is a divisor of $n!$.

This is a version of what's now known as Lagrange's theorem -- it is the version involving transitive group actions.