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 $g_1,g_2\in G$ and then show that $g_1*g_2\in k$,
Id=e. $e\in g$ then $e\in k$,?

2. Originally Posted by mancillaj3
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 $g_1,g_2\in G$ and then show that $g_1*g_2\in k$,
Id=e. $e\in g$ then $e\in k$,?
a subset $K$ of a group is a subgroup if and only if $\forall a,b \in K: \ ab^{-1} \in K.$

3. Originally Posted by NonCommAlg
a subset $K$ of a group is a subgroup if and only if $\forall a,b \in K: \ ab^{-1} \in K.$
And it must be nonempty

4. Originally Posted by NonCommAlg
a subset $K$ of a group is a subgroup if and only if $\forall a,b \in K: \ ab^{-1} \in K.$
Thanks for this very useful criteria to prove a set is a sub-group. I always used to work with the axioms of groups.