Question

Let $\mathit{k}$b e a subgroup of a group$\mathit{G}$ and let $\mathit{a}\mathbf{\in }\mathit{G}$. Prove thatrole="math" localid="1654334784177" $\mathit{k}\mathit{a}\mathbf{=}\mathit{k}$ if and only if $\mathit{a}\mathbf{\in }\mathit{k}$.

Short answer

It is proved that $ka=k$.

Step-by-step solution

Determine Forward direction

Consider the given functions, $k$is a subgroup of $G$, and assume that $a\in G$.

Suppose that $ka=k$, as $k$is a subgroup identity $e$in $G$. Then $ea=a\in ka=k$,

Then this is complete proof of forward direction.

Determine ka=k

Now taking an element $x\in ka$, then,

$x=ka$for some $k\in k$.

As both role="math" localid="1654335789967" $a$and role="math" localid="1654335793500" $k$are in role="math" localid="1654335797063" $k$. Here role="math" localid="1654335800461" $k$is subgroup.

It follows that role="math" localid="1654335860212" $ka\in k$is verifying that $ka\subset k$.

Now taking an element role="math" localid="1654336021704" $k\in k$, asrole="math" localid="1654336017958" $a\in k$and $k$is subgroup, role="math" localid="1654336026517" $k{a}^{-1}\in k$, thusrole="math" localid="1654336041429" $k=k{a}^{-1}a\in ka$.

And this is verifying that $ka\subset k$.

Hence $ka=k$.