Finite group theory - validity of the following conjecture

Asked Mar 15 '17 at 21:42

Active Mar 15 '17 at 21:44

Viewed 51 times

Wonder whether the following conjecture holds:

Let: $(G,*)$ - finite group, in which every proper subgroup is cyclic. Then: $(G,*)$ is cyclic.

Intuitively, there are no trivial counterexamples. However, I may miss something. Any help very appreciated!

edited Mar 15 '17 at 21:43

asked Mar 15 '17 at 21:42

3

Doesn't even have to be abelian...$S_3$ is an example. – lulu Mar 15 '17 at 21:45
$S_3$ has a cyclic subgroup of order 3, and several subgroups of order 2. – Doug M Mar 15 '17 at 21:46
2

If this was true then it would follow immediately by induction that all finite groups were cyclic. – Derek Holt Mar 16 '17 at 00:27

0 Answers0