- #1

Dedrosnute

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## Homework Statement

I need to show that for a group G of order 34 that if the order of the automorphism group is less than or equal to to 33, then G is cyclic.

## Homework Equations

none

## The Attempt at a Solution

I'm mainly trying to do a proof by contradiction. First I assumed that G had <= 33 automorphisms and then supposed that G was not cyclic. So then I tried to construct a group of that sort.

If G has order 34, but G is not cyclic, i.e. generated by an element of order 34, then G must be generated by an element x of order 2 and an element y of order 17. Is this correct? It seems like G would still be cyclic of order 34 and would be generated by the element xy.

Anyhow, I tried constructing the possible automorphisms. There would be an automorphism mapping y to each power of y between 1 and 17, so that is 17 automorphisms. Then I wasn't sure how to use the factor of 2. If I could use the factor of 2, I might be able to double the amount of automorphisms, but it seems like x could only be mapped to x, since x = x^-1 is the only element of order 2.

Could someone help me out please?