Algebraic fixed point theorem

Question

I was wondering if there are some "algebraic" fixed point theorems, in group theory.

More precisely, given a group $G$ and a group morphism $f : G \to G$, what conditions on $G$ and $f$ should we demand, so that $f$ has a non-trivial fixed point (i.e. $\exists x \neq 1_G, f(x)=x$) ?

Here are my thoughts :

1. This « non-trivial fixed point condition » is sometimes a strong condition. For instance, if $G = \mathbb Z$, then the only $f \in \text{Hom}(G,G)$ to have a non-trivial fixed point is the identity.

2. The set of fixed point $\{y \in G \mid f(y)=y\}$ is a subgroup of $G$.

3. Let $G = \mathbb Z / n\mathbb Z$. Assume that $n=ab$ with $a,b>1$. If $f([1]_n) = [a+1]_n$, then $f$ has a non trivial fixed point, namely $x=[b]_n$.
4. This question may be «artificial» ; I don't know if a morphism with a non trivial fixed point can be useful in other contexts...
5. I don't see a natural way to turn this problem into a « group action » problem (to get some results about fixed points). I tried $G \curvearrowright \text{Im}(f)$ by defining $g \bullet f(x) := f(g)f(x) = f(gx)$, but this doesn't seem to help...

Thank you in advance !

Answer 1

The only fixed point theorem involving finite groups I know is the following:

$p$-group fixed point theorem: Let $G$ be a finite $p$-group acting on a finite set $X$. Then $|X^G| \cong |X| \bmod p$. In particular, if $|X| \not \equiv 0 \bmod p$, then $G$ has a fixed point.

For example, applied to the conjugacy action of a finite $p$-group on itself, we conclude that such a group has nontrivial center. We can get a statement of your form by asking that $G$ is a $p$-group and $f$ has order a power of $p$.

Other applications are given here.