The question is related to Taft's Theorem about G-invariant radical complements. Let $A$ be an associative unitary finite-dimensional $K$-Algebra posessing a separable factor Algebra by ist nilradical. Suppose $G$ is a finite group such that the order of $G$ is not divisible by the characteristic of $K$ and $G$ is acting on $A$ as auto- or anti-automorphism. Taft has proven that a $G$-invariant radical complement exists. A $G$-invariant radical complement is a fix point of the set of all radical complements under the Action of $G$.


If we assume a finite field, then one can prove that the number of radical complements is a power of the characteristic of $K$. A natural question is:


Let $G$ be a finite $p'$-Group acting on a set $M$ of $p$-power order. Is there a fix Point in $M$ under $G$?


Within $S\_3$ one can construct examples of a subgroup of order 2 acting on the cosets of $S\_3$ Modulo another subgroup of order 2 -- which is a set containg 3 elements. Within this example no fix Point exist.


My questions are: 


1.) What is so Special within the mentioned situation that a fix point exists.


2.) Is there a General result on fix Points related to the described context?



Operation of a p'-group on a set of p-power order and fix points
I think the guess in your comment is correct: There is the following result of Glauberman:



> 
> **Theorem** (Glauberman). Let the finite group $G$ act on the finite group $N$ by automorphisms, where $(\lvert G \rvert, \lvert N \rvert ) = 1$. Let both $G$ and $N$ on a set $\Omega$, where the action of $N$ is transitive, and the compability condition 
>  $$ (\omega n) g = (\omega g) n^g \quad \text{for all } 
>  \omega\in \Omega,\, n\in N,\, g\in G $$
>  holds. Then $\Omega$ has a fixed point under $G$. Moreover, $C\_N(G)$ acts transitively on the $G$-fixed points. 
> 
> 
> 


In your situation, this lemma applies with $N = 1 + \operatorname{rad}(A)$, which acts transitively on the set of complements of the radical, if I understand correctly. 


Glauberman's result appears as Lemma 3.24 in Isaacs's text on finite groups or in Glauberman's original paper (Theorem 4).


*Isaacs, I. Martin*, Finite group theory. Graduate Studies in Mathematics 92. AMS 2008. [ZBL1169.20001](https://zbmath.org/?q=an:1169.20001).  

*Glauberman, George*, [**Fixed points in groups with operator groups**](http://dx.doi.org/10.1007/BF01117119), Math. Z. 84, 120-125 (1964). [ZBL0123.02601](https://zbmath.org/?q=an:0123.02601).


