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Cyclic Groups 
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#1
Aug1809, 11:15 AM

P: 1,133

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


#2
Aug1809, 11:44 AM

P: 1,059

A cyclic group is generated by a single element.



#3
Aug1809, 01:15 PM

P: 1,133

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



#4
Aug1809, 01:44 PM

P: 286

Cyclic Groups
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 nonnegative integer n so that a^{n}=b.
In order to show a group is cyclic, one must find such a member a. To show it is noncyclic, 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 


#5
Aug1809, 02:47 PM

P: 1,133

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


#6
Aug1809, 09:51 PM

P: 6

The group G that you've presented is certainly noncyclic. Here is a proof: For any element g in G, g^{2}=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! 


#7
Aug1909, 09:39 AM

P: 1,133




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