1. The problem statement, all variables and given/known data 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. 2. Relevant equations none 3. 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?