Abstract We study finite groups G such that the maximum length of an orbit of the natural action of the automorphism group Aut ⁢ ( G ) {\mathrm{Aut}(G)} on G is bounded from above by a constant. Our main results are the following: Firstly, a finite group G only admits Aut ⁢ ( G ) {\mathrm{Aut}(G)} -orbits of length at most 3 if and only if G is cyclic of one of the orders 1, 2, 3, 4 or 6, or G is the Klein four group or the symmetric group of degree 3. Secondly, there are infinitely many finite (2-)groups G such that the maximum length of an Aut ⁢ ( G ) {\mathrm{Aut}(G)} -orbit on G is 8. Thirdly, the order of a d-generated finite group G such that G only admits Aut ⁢ ( G ) {\mathrm{Aut}(G)} -orbits of length at most c is explicitly bounded from above in terms of c and d. Fourthly, a finite group G such that all Aut ⁢ ( G ) {\mathrm{Aut}(G)} -orbits on G are of length at most 23 is solvable.

中文翻译：

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只有小的自同构轨道的有限群

摘要 我们研究了有限群 G，使得自同构群 Aut ⁢ ( G ) {\mathrm{Aut}(G)} 在 G 上的自然作用的轨道的最大长度由一个常数从上方限定。我们的主要结果如下：首先，有限群 G 只承认 Aut ⁢ ( G ) {\mathrm{Aut}(G)} - 长度最多为 3 的轨道当且仅当 G 是阶数之一的循环, 2, 3, 4 or 6, or G 是克莱因四群或3次对称群。 其次，有无穷多个有限(2-)群G，使得Aut ⁢ ( G ) { \mathrm{Aut}(G)} -orbit 在 G 上是 8。第三，d 生成的有限群 G 的阶使得 G 只承认 Aut ⁢ ( G ) {\mathrm{Aut}(G)} -orbits根据 c 和 d，长度至多为 c 的 是从上面明确限定的。第四，