Question

Let E be the group of even integers and N the subgroup of all multiplies of 8.

To what well-known group is $\mathit{E}\mathbf{/}\mathit{N}$ isomorphic.

Short answer

$E/N$is isomorphic to ${\mathrm{\mathbb{Z}}}_4$.

Step-by-step solution

Step 1:Theorem 8.8

Every group of order 4 is isomorphic to either${\mathbf{Z}}_{\mathbf{4}}$or${\mathbf{Z}}_{\mathbf{2}}\mathbf{\times }{\mathbf{Z}}_{\mathbf{4}}$ .

Showing that E/N  is isomorphic to  Z4.

Since $E/N=\{0,2,4,6\}$, its order is 4.

Therefore, from the theorem, it should be isomorphic to ${\mathbb{Z}}_4$ or ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_4$.

As we know,

${\mathrm{\mathbb{Z}}}_4=\{0,1,2,3\}$

${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_4=(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3).$

Therefore, it can be seen that $E/N$is isomorphic to ${\mathrm{\mathbb{Z}}}_4$, with the generator 2.