Let and let be the order of the largest cyclic subgroup of G. What is ? And what would be the number of distinct cyclic subgroups of order in ?

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- August 11th 2010, 10:25 PMdemodeAutomorphism
Let and let be the order of the largest cyclic subgroup of G. What is ? And what would be the number of distinct cyclic subgroups of order in ?

- August 11th 2010, 11:00 PMSwlabr
You should start by thinking about what this automorphism group looks like. Essentially, this comes down to `where can the generator, 1, be sent to by an automorphism?'. (e.g. is an automorphism. Can you think why this is?)

So, where can the generator be sent to? - August 14th 2010, 04:21 AMdemode
The generator is sent to , as is isomorphic to ?

I think the order of the largest cyclic subgroup of G is

For an element of we know , we'll know fora any , because

I think the property that isomorphisms preserve orders would tell us that , then I think members of would be the candidates for .

The largest member of is 314. But I don't know if this helps... - August 14th 2010, 11:39 PMdemode
Oh, I meant since is an automorphism we have for some and any .

But how does this help me to prove the question? - August 16th 2010, 12:47 AMSwlabr
Yes, you need to find out where you can send the generator too. Your choices are precisely those numbers which generate . So, which elements of your group generate it? How many of them are there?

This will give you the order of your Automorphism group (why?). - August 16th 2010, 04:59 AMdemode
Thanks for the reply. Since , , where

Therefore the generators of are the elements

Is this right? If so how do I know how many elements there are in ? (there are too many to count) - August 21st 2010, 01:25 PMdemode
Okay I know that if G is a cyclic group of order n, then .

In our case if we have

Does this mean the order of the largest cyclic subgroup of G is 63? Because .

Am I right? - August 22nd 2010, 11:39 PMSwlabr