Prove that the intersection of two subgroups is a subgroup. In more detail, if  $G$  is a group and $H_1$, $H_2$   are subgroups of G   then $H_1 \cap H_2$   is a subgroup of  G.
Next, give an example of a particular group $G$  (any one you like), and two different subgroups  $H_1$, $H_2$ of $G$  , compute the intersection $H_1 \cap H_2$  , and verify it is indeed a subgroup. 
Finally, give three examples showing that $H_1 \cup H_2$   need not be a subgroup of $G$ .

You can see the first from the definition of subgroup or the subgroup criterion quite easily. It's quite easy to make $H_1\cup H_2$ not a group... Consider two subgroups that intersect trivially. What happens when you add non-identity elements from these two subgroups?
