I am investigating applications of the Hurwitz quaternions: https://en.wikipedia.org/wiki/Hurwitz_quaternion $$H_u = \left\{a+bi+cj+dk\mid (a,b,c,d) \in\mathbb{Z}\mbox{ or }(a,b,c,d) \in\mathbb{Z}+\frac{1}{2}\right\}$$
As an additive group, H is free abelian with generators $\dfrac{1 + i + j + k}{2}, i, j, k$
I'm trying to define modulo and square root for the Hurwitz quaternions, so I can express a large Hurwitz prime as a congruence of two smaller integers: $p \equiv 3 \text{ mod } 4$ where p is a Hurwitz prime, similar to the Rabin cryptosystem
One way I know of doing this is by knowing that $H$ forms a cyclic group. I am not sure how to determine that or prove it, as my math knowledge is limited. Any help or guidance is appreciated!
No, this group is not cyclic as it is free abelian of rank $4$. A free abelian group is cyclic if and only if the rank is $1$, i.e., if it has a basis with $1$ element.
Moreover the unit group of the Hurwitz quaternions is a split extension of the quaternion group $Q_8$ and the cyclic group $C_3$ and hence also not cyclic, see here.