We have a group (G,+) and his semigroup (H,+) then there is subset K $\subseteq$ H and K is semigroup of H: for any x,y in K: x+y in K . Where "+" is any operation.
What is defference between semigroup and group - semi-one doesn't have Identity or/and Inverses element.
(1)Identity element - $\exists e \in K, \forall a \in K: e+a = a+e = a$
(2)Inverses element - $ \forall a \in K, \exists b \in K,: a+b = b+a = e$
Let G is torsion group, so for any x in G: |x| < $\infty$. Then $ \forall a \in G, \exists n(a) = n \in \mathbb{N}: a^n = e \in G$ and $e \in K$ (1)
(2) $\forall a \in K, \exists n(a) = n \in \mathbb{N}: a^{n-1} + a = a + a^{n-1} = e$
So condition is G is a torsion group. Is it enough?
Your question is a little bit unclear. I think you are asking for necessary and sufficient conditions on a group $(G, +)$ which ensure that every non-empty set $K$ of $G$ that is closed under $+$ (i.e., any subsemigroup of $(G, +)$ viewed as a semigroup) is a subgroup.
You have shown that if $G$ is a torsion group then any subsemigroup $K$ will contain the identity and the inverse of each of its elements and so will be a subgroup. This shows that being a torsion group is a sufficient condition for the property of interest.
You also need to show necessity. I.e., that any group all of whose subsemigroups are subgroups is a torsion group. To see this, assume that $G$ is not a torsion group. Then there is a $g \in G$ of infinite order, i.e., such that $g^n \neq e$ for any positive $n$. But then if you take $K = \{g, g^2, g^3, \ldots\}$, $K$ is a subsemigroup of $G$ that is not a subgroup. This shows that the being a torsion group is a necessary condition for the property of interest.