# Group acting coprimely by automorphism

Consider that a group $$A$$ acts by automorphism on a finite group $$G$$. If this action is coprime, i.e. $$\gcd(|A|,|G|)=1,$$ can we affirm that this action is fixed point free, i.e. $$C_G(A)=1$$?

I tried to think about the order of the automorphism and the order of na element of $$G$$, but it doesn’t work.

• What does "coprime" mean here? Does it mean $\gcd(|A|, |G|) = 1$? – Qiaochu Yuan Sep 29 at 18:41
• What is a coprime action? – JCAA Sep 29 at 18:42
• If the action is by automorphisms, the identity will always be a fixed point – ahulpke Sep 29 at 18:54
• @ahulpke I think a fixed point free action of a group is usually defined to mean one in which only the identity is fixed i.e. $C_G(A) = 1$, and I am guessing that coprime means $(|A|,|G|)=1$. But the answer to the question is clearly no, coprime actions need not be fixed point free. – Derek Holt Sep 29 at 19:03
• Yes, @QiaochuYuan, coprime action mean gdc(|A|,|G|) = 1 – Eliana C. Rodrigues Sep 29 at 19:03

As Derek Holt pointed out in the comments, the action may have fixed points, even if we don't count the identity. For example, take $$G$$ to be the quaternion group of order 8, and let $$A = \mathbb Z/3\mathbb Z$$ act by cycling $$i, j$$, and $$k$$. This fixes the central element $$-1$$.