On Minimal Orders of Groups with Odd Order Automorphism Groups

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Let G be a finite group. Suppose that the number of prime factors of the order of G, counting repetitions, is less than or equal to five and that the order of G is not equal to p 5 for some prime number p > 3. Then we show that the order of the automorphism group of G is even. We finish with some conjectures on the smallest groups with certain odd order automorphism groups.