Examples of classes $\mathcal{C}$ of structures such that every finite group is isomorphic to the automorphism group of a structure in $\mathcal{C}$

Solution 1:

The classes of monoids and semigroups are universal.

Adjoining an identity to a semigroup doesn't change the automorphism group, so it's enough to prove this for semigroups, for which I'll use the fact that the class of directed (acyclic) graphs is universal.

Let $G$ be a directed graph with vertex and edge sets $V$ and $E$. Define a semigroup $S=V\cup E\cup\{0\}$ with all products equal to $0$ except that, for $v\in V$ and $e\in E$, $v^2=v$, $ve=e$ if $v$ is the initial vertex of $e$, and $ev=e$ if $v$ is the terminal vertex of $e$.

Then it is easy to see that the automorphism group of $S$ is the same as that of $G$.

In fact, since universality of directed graphs only requires finite graphs, even the classes of finite monoids and semigroups are universal.