Polynomial satisfying p(x)=3^{x} for x\in\mathbb{N}

Prove that in any group, an element and its inverse have the same order.

Show that an intersection of normal subgroups of a group G is again a normal subgroup of G

Let ${\displaystyle H\le G}$
Prove
${\displaystyle x^{-1}{y}^{-1}xy\in H\mathrm{\forall }x}$
$y\in G\iff H⊴G$
and
${\displaystyle \frac{G}{H}}$ is abelian.

Prove the following.
(1) $Z\times 5$ is a cyclic group.
(2) $Z\times 8$ is not a cyclic group.

Finding the Units in the Ring ${\displaystyle \mathbb{Z}\left[t\right]\left[\sqrt{t^2-1}\right]}$

Use the isomorphism theorem to determine the group ${\displaystyle G{L}_2\frac{\mathbb{R}}{S}{L}_2\left(\mathbb{R}\right)}$. Here ${\displaystyle G{L}_2\left(\mathbb{R}\right)}$ is the group of ${\displaystyle 2\times 2}$ matrices with determinant not equal to 0, and ${\displaystyle S{L}_2\left(\mathbb{R}\right)}$ is the group of ${\displaystyle 2\times 2}$ matrices with determinant 1. In the first part of the problem, I proved that ${\displaystyle S{L}_2\left(\mathbb{R}\right)}$ is a normal subgroup of ${\displaystyle G{L}_2\left(\mathbb{R}\right)}$. Now it wants me to use the isomorphism theorem. I tried using
${\displaystyle \left|G{L}_2\frac{\mathbb{R}}{S}{L}_2\left(\mathbb{R}\right)\right|=\frac{\left|G{L}_2\left(\mathbb{R}\right)\right|}{\left|S{L}_2\left(\mathbb{R}\right)\right|},}$
but since both groups have infinite order, I don't think I can use this here.

A question on Hamilton Quaternions
How does one prove that ring of Hamilton Quaternions with coefficients coming from the field ${\displaystyle \frac{\mathbb{Z}}{p}\mathbb{Z}}$ is not a divison ring.