Let $H$ be a subgroup of a finite group $G$. Then $[G:H]=\frac{|G|}{|H|}$ is the number of distinct left cosets of $H$ in $G$.
Proof: The group $G$ is partitioned into $[G:H]$ distinct left cosets $g_{1}H,g_{2}H,...,g_{[G:H]}H$. Then $|g_{i}H|=|H|$ for all $i$. Then $G=g_{1}H\cup g_{2}H\cup \cdots \cup g_{[G:H]}H$. Then $|G|=|g_{1}H|+|g_{2}H|+\cdots +|g_{[G:H]}H|=|H|+|H|+\cdots +|H|$ ($[G:H]$ times) $=[G:H]|H|$. $\square$
I just want to know whether I can write $g_{[G:H]}H$ and it makes sense?
$[G:H]$ is a natural number like any other, so if the convention is that of $g$'s enumerated in underscript, then $g_{[G:H]}$ makes just as much sense as $g_3$, and so do all notations which derive from it.