cyclic group associated with chaotic behavior of Primes

The Primes Modulo 10 give endings:
{1,3,7,9}
The multiplication group of {1,I,-I,-1}
which it called a four element cyclic group ( in this order , not the usual order used in this group )
behaves like the Primes modulo 10
in multiplication.
If you use this group to make an IFS ( with ration 1/2)
you get a space filled square.
We don't know for sure ( no one has proved)
that the appearance of the primes by modulo ending is " normal",
but experiments up to several billion seem to indicate that they are
like the digits of Pi random in occurrence.
Artur Jazinski noticed the similarity of the groups and pointed it out to me.
He and I have done some recent work on quotient groups/ rings of polynomials
like the Theta1 minimal Pisot ( also a four element group).

Essentially what this says is that if the primes appear randomly
they have an integer topological dimension... Gaussian normal s=2,
but if they are less than s=2,
then they have a fractal Brownian modulo ten digit appearance.

Mathematica code to get from primes to the cyclic group:
(* {1,3,7,9}->{1,I,-I,-1}
f[n_] = If[Mod[Prime[n], 10] - 1 == 0, 2, If[Mod[Prime[n], 10] - 3 == 0, 1/2,
If[Mod[Prime[n], 10] - 1 == 0, 3/2, 1]]]
g[n_] := (-1)^f[n]
Table[{g[n], Mod[Prime[n], 10]}, {n, 4, 10}]
Table[g[n]*g[m], {n, 4, 10}, {m, 4, 10}]
.