Exercise
How many groups of order 15 are there up to isomorphism?
0
1
2
3
No.
There is the cyclic group of order 15.
Correct!
Do you know why? There are elements of order 3 and 5 and they must commute
(for otherwise, there would be at least 3 subgroups of order 5 and at least 5 subgroups of order 2,
making up at least 3×4+5×2 = 22 elements, contradicting that there are at most 15 elements).
But then the group is isomorphic to Z/3Z×Z/5Z,
whence to Z/15Z.
No. By the theorem there are elements of order 3 and 5. Show that they must commute.
No. By the theorem there are elements of order 3 and 5. Show that they must commute.