$Ka=K$ iff $a\in K$ Let $K$ be a subgroup of of a group $G$. Let $a \in G$ 
Prove $Ka=K$ iff $a\in K$

need to show $Ka=K \Rightarrow a \in K $ and $a\in K \Rightarrow Ka=K$
$\Rightarrow]$($Ka=K \Rightarrow a \in K $)  (no clue appreciate a hint )
$\Leftarrow]$  ($a\in K \Rightarrow Ka=K$ (same dont know how to approciate)

Def $$ Ka =\{ka: k\in K \}$$

$K$ is a subgroup and $e\in K, Ka=K$ implies $ea=a\in K$.
On the other hand if $a\in K, a^{-1}\in K, x\in K$ implies $x=(xa^{-1})a, xa^{-1}\in K$ since $x\in K, a^{-1}\in K$ so $x=(xa^{-1})a\in K$. This implies that $K\subset Ka, Ka\subset K$ since a group is stable by multiplication.
