Comment by Gro-Tsen on Existence of a special ordering of the elements of a finite group (II)

I'm not too familiar with this, but if you consider the product of the $1+g+\cdots+g^{\textrm{ord}(g)-1}$ not over all elements of $G$ (in some order) but only over some of them (but still want the product to be the sum of all elements of $G$ in the group algebra), this seems related to the problem of finding cyclic subgroups $\langle g_i\rangle$ of $G$ such that each element of $G$ is expressible uniquely as product $\prod g_i^{r_i}$ (in a fixed order). This, in turn is related to “minimal logarithmic signatures” in cryptography, see, e.g. here.