1.
Write down all subgroups of \(\Z_{12}\text{.}\) Think about how you know you are not missing any.
Worksheet Generated Subgroups
The group \(\Z_{12}\) contains 12 elements. The group \(S_5\) contains 120 elements. Thus there are \(2^{12} = 4096\) subsets of \(\Z_{12}\) and \(2^{120} \approx 1.3 \times 10^{36}\) subsets of \(S_5\text{.}\)
How many of these are subgroups?
2.
If you know that a subgroup of \(\Z_{12}\) contains the element \(4\text{,}\) what else must it contain? In other words, what is the smallest subgroup of \(\Z_{12}\) that contains \(4\text{?}\) We call this subgroup \(\langle 4 \rangle\text{.}\)
3.
Which of the subgroups of \(\Z_{12}\) can be written as \(\langle n \rangle\) for some \(n\text{?}\) That is, which subgroups can be generated by a single element? Such subgroups are called cyclic.
4.
Now let’s look for subgroups of \(S_5\text{.}\) Find at least 5 different subgroups of \(S_5\text{.}\)
5.
Is every subgroup you found generated by a single element? Can you find a subgroup that is not generated by a single element?
