Question

If $\mathit{G}$ is the additive group $\mathbf{\square }\mathbf{/}\mathbf{\square }$, what are the elements of the subgroup ? $\mathit{G}\mathbf{\left(}\mathbf{2}\mathbf{\right)}$Of role="math" localid="1657360650385" $\mathit{G}\mathbf{\left(}\mathit{p}\mathbf{\right)}$ for any positive prime $\mathit{p}$?

Short answer

The elements in the subgroup is$G\left(2\right)=\left\{\left[\frac{a}{2^n}\right]\in \square /\square |n\in N,2⟩a\right\}$ and$G\left(p\right)=\left\{\left[\frac{a}{p^n}\right]\in \square /\square |n\in N,p|a\right\}$ .

Step-by-step solution

Elements of subgroup G(2)

Let G be the additive group $\square /\square$ then the elements in the subgroup $G\left(2\right)$will be

$G\left(2\right)=\left\{\left[\frac{a}{b}\right]\in \square /\square \mid \exists n\in N:\left[\frac{2^{n}a}{b}\right]=\left[0\right]\right\}$ .

Here, $\left[\frac{a}{b}\right]\in \square /\square$if and only if there exists some $n\in N$ such that $b\mid 2^{n}a$.

If$(a,b)=1$ then there exists $n\in N$ such that $b=2^n$.

Therefore, $G\left(2\right)$ can be written as $G\left(2\right)=\left\{\left[\frac{a}{2^n}\right]\in \square /\square |n\in N,2⟩a\right\}$.

The subgroup G(p)   

The subgroup $G\left(2\right)$ is $G\left(2\right)=\left\{\left[\frac{a}{2^n}\right]\in \square /\square |n\in N,2⟩a\right\}$. Similarly, for any positive prime p, the subgroup $G\left(p\right)$ will be $G\left(p\right)=\left\{\left[\frac{a}{p^n}\right]\in \square /\square |n\in N,p|a\right\}$.

Therefore, $G\left(p\right)=\left\{\left[\frac{a}{p^n}\right]\in \square /\square |n\in N,p|a\right\}$.