If we say that G is a group under *, and k is assume to be a subset, in order to prove that k is a subgroup, we need to find the closure, ID, and inverse.....

My quetion for closure is, do we let for example the elements $\displaystyle g_1,g_2\in G$ and then show that $\displaystyle g_1*g_2\in k$,

Id=e. $\displaystyle e\in g$ then $\displaystyle e\in k$,?