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 $G$ .
Best Answer
Use the definition of a subgroup. You know that they both share the identity element, and for any element they share, they must also share the inverse. Can you take it from here?