Basic Subgroup Conditions Problem 1:
Prove that a subset $H$ of $G$ is a subgroup if and only if it satisfies the following conditions.


*

*The identity $e$ of $G$ is in $H$.

*If $h_1, h_3 \in H$, then $h_1h_2 \in H$.

*If $h \in H$, then $h^{-1} \in H$.


Problem 2:
Let $H$ be a subset of a group $G$. Then $H$ is a subgroup of $G$ if and only if $H \neq \emptyset$, and whenever $g, h \in H$ then $gh^{-1}$ is in $H$.
I'm not really sure where to start on these problems - I am just starting to learn abstract algebra on my own.
 A: I assume that your definition of "subgroup" is "a subset which is a group under the operation of $G$."
For problem 1, you just need to show that the conditions imply that $H$ satisfies the axioms of being a group; you will need part 2 to show that multiplication restricts to an operation on $H$; then use part 1 to get that $H$ has an identity; and part 3 gives you the inverses. The converse (that if it is a subgroup then it satisfies these conditions) should be easy.
For problem 2, showing that if $H$ is a subgroup then these two conditions hold should be easy. To show that the conditions imply $H$ is a subgroup, try to show that it satisfies the three conditions in problem 1. Since $H$ is nonempty, there is an element $x\in H$. Now apply the condition with $g=h=x$ (note that we do not require $g$ and $h$ to be distinct!) to conclude that $H$ contains the identity. Then take $h\in H$, and set $g=e$ to conclude that if $h\in H$ then $h^{-1}\in H$. And finally, given $g,h\in H$, use $g$ and $h^{-1}$ to show that $H$ is closed under products. 
