Let $G$ be a group and $m \in G$, $K$ be a subgroup and $m * m \in K$. Is it always true that $m \in K$?

Question

I'm trying to use this in another proof, but I think it might be false and I'm out of ideas to prove it.

Answer 1

No way. In the group $\mathbb{Z}$ consider the subgroup $E$ of evens. Then $1+1 \in E$ but $1 \notin E$.

Answer 2

Hint $G=\mathbb Z, K=2 \mathbb Z$.

Answer 3

No: Let $K=\mathbb{Q}^*$ and $G = \mathbb{R}^*$. Then $\sqrt{2}*\sqrt{2}\in K$ but $\sqrt{2}\notin K$.

Answer 4

Not necessarily.

Consider the rationals as a subgroup of real numbers.

$\sqrt 5 \times \sqrt 5 $ is in the subgroup but $ \sqrt 5$ is not.