Coprime automorphism group implies cyclic with order a cyclicity-forcing number
- is the trivial group.
- is isomorphic to the cyclic group of order equal to , where the are pairwise distinct primes, and does not divide for any .
- We first show that the group must be Abelian, otherwise the order of the inner automorphism group would divide the order of the group as well as of its automorphism group.
- Next, we show that for every prime divisor of the order of the group, the -Sylow subgroup must be cyclic of order . Thus, the whole group is cyclic of order where the are distinct primes.
- Finally, we observe that the automorphism group of a group of this form has order . For this to be relatively prime to we need to impose the additional condition that does not divide for any .