Cyclic Groups

[Gear300]

Main Question or Discussion Point

The problem is to verify that {(1), (1 2), (3 4), (1 2)(3 4)} is an Abelian, noncyclic subgroup of S₄.

I was able to show that it is Abelian through pairing the permutations, but my mind stopped at the noncyclic part. When showing that a group is cyclic or noncyclic, what exactly do I have to show?

 

[robert Ihnot]

A cyclic group is generated by a single element.

 

[Gear300]

Therefore, any one of those elements should be able to generate the others, right?

 

[Elucidus]

Yes, if it were cyclic. In order for a group to be cyclic then there must exist a member a so that for all members b, there exists a non-negative integer n so that aⁿ=b.

In order to show a group is cyclic, one must find such a member a. To show it is non-cyclic, one must show that there is a member b which cannot be the power of any other member (it is obviously the 1^st power of itself).

I'd look at (1 2)(3 4) and see if one can show whether it is a power of any of the others.

--Elucidus

 

[Gear300]

Doesn't seem as though (1 2)(3 4) is a power of any of the other elements.
Does n have to be non-negative (in order for it to be a group, shouldn't n also be inclusive of negative integers - to identify the inverses)?

 

[aziz113]

The group G that you've presented is certainly noncyclic. Here is a proof: For any element g in G, g²=1. However, the order of the group is 4, and so no single element can generate the group. Thus the group is not cyclic.

Hope that helps!

 

[Gear300]

The group G that you've presented is certainly noncyclic. Here is a proof: For any element g in G, g²=1. However, the order of the group is 4, and so no single element can generate the group. Thus the group is not cyclic.

Hope that helps!

Thanks for the help.