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There is a theorem by Wang and Chen that says: when the finite group $A$ acts via automorphisms on the finite group $G$ with $|A|$ and $|G|$ coprime, and $C_G(A)$ is either odd-order or nilpotent, then $G$ is solvable. The proof uses the classification of finite simple groups.

A precise reference is given at this Google Scholar link. It's cited dozens of times. But there seems to be no online copy of this paper, and the journal it was published in has ceased running and has no online issues.

Does anyone know where to get a copy, or any other reference that goes through the proof?

EDIT: The full reference is:

Yan Ming Wang and Zhong Mu Chen, Solubility of finite groups admitting a coprime order operator group, Boll. Un. Mat. Ital. A (7) 7 (1993), no. 3, 325–331

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    $\begingroup$ My school library says it has a copy of Boll. Un. Mat. Ital. A (7). I can send a scanning request if you need. $\endgroup$ Commented Oct 4 at 20:59
  • $\begingroup$ @user2249675: that would be most appreciated! $\endgroup$ Commented Oct 4 at 22:01
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    $\begingroup$ I sent it but you may have to wait until Monday. $\endgroup$ Commented Oct 4 at 22:30
  • $\begingroup$ The library said it did not find the said article. :( $\endgroup$ Commented Oct 6 at 15:17
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    $\begingroup$ I just sent it. $\endgroup$ Commented Oct 6 at 20:44

2 Answers 2

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I obtained a copy of the article in question, but due to copyright I cannot post it publicly. If anyone wants a copy of it, leave a comment below and I'll find a way to send it to you.

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  • $\begingroup$ @Deif Please leave an email. You can delete the comment after receiving it. $\endgroup$ Commented Oct 7 at 0:50
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    $\begingroup$ @Deif I just sent it. $\endgroup$ Commented Oct 7 at 1:44
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I don't have access to the article in question, but going through various related papers, I think we should be able to reconstruct a proof.

Basically, I think you can argue as in the proof of Theorem 2.1 in Beltrán, Antonio; Actions with nilpotent fixed point subgroup. Arch. Math. (Basel) 69 (1997), no. 3, 177--184.

First, by the coprime assumption:

Lemma: If $N \trianglelefteq G$ is $A$-invariant, then $C_{G/N}(A) = C_G(A)N/N$.

Proof: See Theorem 1 in "Glauberman, George; Fixed points in groups with operator groups. Math. Z. 84 (1964), 120-125".

So proceeding by induction we may as well assume that $G$ is minimal normal in $G \rtimes A$. And so $G = H_1 \times \cdots \times H_k$, with $H_i$ pairwise isomorphic simple groups, and by contradiction assume that the $H_i$ are nonabelian.

Then as in the paper by Beltrán, reduce to the case where $G$ is simple, so by coprime assumption $G$ is of Lie type. And moreover $A$ must act by field automorphisms, so $G = X(q^r)$ and $C_G(A) = X(q)$ for some Lie type $X$. But then $X(q)$ is not of odd order, nor nilpotent.

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