Is Lagrange's theorem the most basic result in finite group theory?

Consider the representation of $\langle a \rangle$ on the free vector space on $G$ induced by left multiplication. Its character is $|G|$ at the identity and $0$ everywhere else. Thus it contains $|G|/|\langle a \rangle|$ copies of the trivial representation. Since this must be an integer, $|\langle a \rangle|$ divides $|G|$. Developing character theory without using Lagrange's theorem is left as an exercise to the reader.

I am late... Here is a proposal, probably not far from Ihf's answer.
For $a \in G$ of order $p$, define the binary relation $x\cal Ry$ : $\exists k\in \mathbb{N} ; k<p$ such that $y=a^kx$
$\cal R$ is an equivalence relation on $G$ and sets up a partition of $G$.
A class is defined by $C_x=\left\{a^kx|k=0,1,\ldots,p-1 \right\}$
All the classes have $p$ elements, then $n=|G|$ is a multiple of $p$ ; $(n=pm)$ and $a^n=a^{pm}=e$