Question

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

1. Show that$\mathbf{E}\mathbf{/}\mathbf{N}$ has order 4.

Short answer

$E/N$ has order 4.

Step-by-step solution

As given in the question

$\mathrm{E}=\{0,2,4..........\}$

$\mathrm{N}\subseteq \{0,8,16........\}$

Showing that E/N  has order 4

Since E/N denotes all-right cosets of N in E.

So, elements in the quotient group or factor group of E by N can be given as:

$\mathrm{E}/\mathrm{N}=\{0,2,4,6\}$

Therefore, it can be seen that E/Nhas only 4 elements, i.e., its order is 4.