Suppose $S$ is a set of groups of order $n$, is there a binary operation $*:S\times S \to S$ that is definable on $S$?
The obvious operations I started with were
- Cartesian product, but that produces groups of order $n^2$.
- Intersection, but that produces groups of order $\leq n$.
- Matrix multiplication of the Cayley table representations, but that doesn't produce another $n\times n$ matrix over the integers $\{0,1,...\,n-1\}$
I'm curious, are there are any known operations one can define that takes pairs of groups and produces one of the same order?