I am trying to find a counterexample to a claim about groups.
I need a random generator (a program in any programming language, or an algorithm) which will generate a random group $G$ of any order such that:
- $G$ has a p-subgroup $P$ which is not a Sylow p-subgroup
- $G$ has a normal subgroup $N$ whose order is coprime with $|P|$
Is there any known work on generating random groups in general? The only way I see now is to create a multiplication table, check group axioms, check the properties I need. But this is inefficient. I do not think I can every generate a group satisfying the above properties by simply trying all tables. Any thoughts?
It is easy to write down a group with the desired properties; and, you should be able to see how to generalise this to provide infinitely many more examples. Take $G = C_2^2\times C_3$, and $P = C_2$ and $N = C_3$. The Sylow $2$-subgroup of $G$ is $C_2^2$, $P$ is a non-Sylow $2$-subgroup, and $N$ - with order coprime to $2$ - is certainly normal in $G$ because $G$ is abelian.