# A Question About Coprime Actions

I'm dealing with the following problem in Isaacs Finite Group Theory [4D.3], I would appreciate if you could help:

Problem: Let $$A$$ act via automorphisms on $$G$$ , where $$(\vert G \vert, \vert A \vert ) = 1.$$ Suppose that $$A$$ acts trivially on every $$A$$-invariant proper subgroup of $$G$$, but that the action of $$A$$ on $$G$$ is nontrivial. Show that $$G$$ is $$p$$-group for some prime $$p.$$

My attempt: For a contradiction, assume that $$G$$ is not a $$p$$-group. Using Glauberman Lemma, we can see that there exists an $$A$$-invariant Sylow $$p$$-subgroup of $$G$$ for each prime. Due to hypothesis, for any $$A$$-invariant Sylow $$p$$-subgroup $$P$$, it must be $$[P,A]=1$$. That is each such a Sylow $$p$$-subgroup is contained in $$C_G(A)$$. From the hypothesis again, we know that $$C_G(A) \neq G.$$ How can I get the desired result from this point?

Thanks.

• From the solution-verification tag wiki: "A question with this tag should include an explanation for why the argument presented is not convincing enough." Sep 18 at 18:14
• Thanks, question draft corrected. Sep 18 at 18:30
• You are still misusing the solution-verification tag: you don't present a solution, and you don't ask about a given solution. It's easy here: since you don't have a solution, how can you be asking for a solution verifcation? Sep 18 at 18:41
• For each prime $p$ dividing $|G|$, we know that $A$ centralizes some Sylow $p$-subgroup $G_p$ of $G$, and $G$ is generated by the subgroups $G_p$, so $A$ centralizes $G$, contradiction. Sep 18 at 21:54
• I got it, thanks. Sep 19 at 7:30