As I understand
There can't be a general algorithm to decide if two finite groups are isomorphic, Wikipedia.
But are there efficient algorithms for all subgroups of $S_n$ for say $n=10$ or so?
I accepted the answer since it really was the answer (I misread Wikipedia and became a bit confused) but I would appreciate further suggestions on efficient algorithms for "small" $n$, for my blog on Zet.
In Zet finite groups are represented as sets of permutations. Sets are implemented as bundles on stacks and permutations as vectors where for example $(4,3,2,1,5)$ represent $(1,4)(2,3)$.
I have an idea of a pseudo isomorphi, checking some characteristics as:
- The cardinals of the groups
- The cardinals of the sets of all (cyclic) subgroups
- Sorted vectors of the cardinals of the subgroups above
No, there are definitely algorithms to determine whether two finite groups are isomorphic or not.For a simple inefficient one, just look at all possible bijections and check using a multiplication table whether any of these bijections are homomorphisms.
What is true is that no such algorithm exists for finitely presented groups. See wikipedia.