**Problem 1**. Let be a group and suppose that are two subgroups of Show that if then either or

**Solution**. If or then gives or and we are done. Otherwise, there exist and But then contradiction!

So, as a result, if is a finite group and are two subgroups of with and then That raises this question: how large could get? The following problem answers this question.

**Problem 2**. Let be a finite group and suppose that are two subgroups of such that and Show that

**Solution**. Recall that and thus Hence and so

where and

Now, since and we have and i.e. and So if we let and then and thus

The result now follows from

**Example 1**. The upper bound in Problem 2 cannot be improved, i.e. there exists a group and subgroups of such that An example is the Klein-four group and the subgroups and Then and

**Example 2**. We showed in Problem 1 that a group can never be equal to the union of two of its proper subgroups. But there are groups that are equal to the union of three of their proper subgroups. The smallest example, again, is the Klein-four group