Is Every Group with Only Cyclic Subgroups Itself Cyclic?

[margaret23]

I m having trouble with a couple group theory proofs. I just have no clue how to start. If u could put me on the right path that would be great.

first

prove of disprove that if every subgroup of a group G is cyclic, then G is cyclic.

and second

prove or disprove that every group X of order 6 is communtative.

 

[StatusX]

For the first, I'm assuming G doesn't count as a subgroup of itself. What are your thoughts on whether these are true or not?

 

[margaret23]

the question doesn't state that its not including the whole group as a subgroup, however i think that its safe to assume that they want it proved without.

I' m thinking from what i know about groups that the first one is true.. and that the second one is false.

I think that the second one probably has a counter example somewhat like the quaternions, Q8.

but i don't really know what else to look at.

 

[StatusX]

For the first, remember that every group of prime order is cyclic. So if you had a group of order pq, where p and q are prime, all its proper subgroups must have order p, q, or 1, and so must be cyclic. Are all such groups cyclic?

For the second, can you list all the groups of order 6? There are actually very few of them.

 

[matt grime]

There are actually 2 of them, and since it is trivial to write down two non-isomorphic groups of order 6... (and any proper subgroup of them is cyclic, by the way, by Lagrange's theorem)

 

[margaret23]

Thanks for the help on the second one .. it makes sense now.. for the first question about proving that if all subgroups are cyclic then the group is cyclic. I would think that its true however i do not know how to start the proof. Could you put me on the right track again?? thanks

 

[fourier jr]

first

prove of disprove that if every subgroup of a group G is cyclic, then G is cyclic.

and second

prove or disprove that every group X of order 6 is communtative.

i think you can actually use the second part to prove the first

 

[BSMSMSTMSPHD]

If you have the answer to the second question, then you have a counterexample to the first question.

 

[fourier jr]

ha i beat you to it :tongue:

 

[margaret23]

my counter example to the second one is the diherdral group containing the permentation of 3 elements. i m having trouble rapping my mind around how this is cyclic.. i ve delt a lot with simpler groups like the intergers mod n , i don't quite understand what makes my counter example cyclic if infact it is.. ?

 

[margaret23]

ps.. thanks

 

[BSMSMSTMSPHD]

You are trying to prove or disprove that if every subgroup of a group G is cyclic, then G is cyclic.

Take the group that you found. What do you know about its subgroups?

BTW - This group is not cyclic, as you have said.

 

[margaret23]

Thanks soo much

 

[matt grime]

By definition a group of order n is cyclic if and only if it has an element of order n. The group you have (S_3, or D_3, it's called) has only 6 elements. You can write them all down and work out their orders.