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In what follows, for a group $G$, $\operatorname{Aut}(G)$ and $\operatorname{Out}(G)$ denote the automorphism and outer automorphism group of $G$ respectively, and for $x\in G$, $x^G$ and $x^{\operatorname{Aut}(G)}$ denote the conjugacy class and automorphism orbit of $x$ in $G$ respectively, whereas $\operatorname{C}_G(x)$ denotes the centralizer of $x$ in $G$.

Question:

Is it true that for all nonabelian finite simple groups $S$ and all $x\in S$, one has $|x^{\operatorname{Aut}(S)}|\leq\frac{1}{2}\cdot(|S|-1)$?

Some thoughts/remarks:

  1. Since $|x^{\operatorname{Aut}(S)}|\leq|\operatorname{Out}(S)|\cdot|x^S|$, for the answer to the Question to be "yes" for a given $S$, it is sufficient to have $|x^S|\leq\frac{1}{2|\operatorname{Out}(S)|}\cdot(|S|-1)$, or equivalently, $|\operatorname{C}_S(x)|\geq2\cdot\frac{|S|}{|S|-1}\cdot|\operatorname{Out}(S)|$ for all $x\in S$, so in particular, $|\operatorname{C}_S(x)|\geq\frac{120}{59}\cdot|\operatorname{Out}(S)|=2.033\ldots\cdot|\operatorname{Out}(S)|$ would be sufficient.
  2. Using the above considerations and information from the ATLAS, it can be checked that the answer to the Question is "yes" for all alternating and sporadic $S$.
  3. A Google search for relevant literature has only turned up rather recent lower bounds on the maximum conjugacy class length, such as [2] and [3], and older results on the existence of elements with a large centralizer, such as the ones mentioned in Geoff Robinson's answer to this mathoverflow question.
  4. The Question is motivated by a connection between the existence of large automorphism orbits on a finite group $G$ and the largest Hamming distance a function on $G$ can have from the set of affine functions on $G$ (functions of the form $x\mapsto g\varphi(x)$ for $g\in G$ and $\varphi\in\operatorname{End}(G)$ fixed), see [1, Definition 2.8 and Proposition 2.9] if you are interested.

References:

  1. A. Bors, Worst-case approximability of functions on finite groups by endomorphisms and affine maps, preprint (2017), arXiv:1709.00734.

  2. L. He and W. Shi, The largest lengths of conjugacy classes and the Sylow subgroups of finite groups, Arch. Math. 86:1-6 (2006).

  3. G. Qian and Y. Yang, The largest character degree, conjugacy class size and subgroups of finite groups, to appear in Comm. Algebra, accepted author version.

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