This can be cast as a shortest path problem. Implementations on huge graphs (several million nodes), i.e. larger than the one defined by a dictionary, are very efficient and usually take advantage of both the source and destination rather than just performing breadth first search from the source. I can think of Andrew Goldberg's work, see e.g.
Thanks for the Goldberg pointer. I have a rudimentary bidirectional, parallel BFS working which I'll be blogging about once it's ready. It runs very fast even on huge graphs.
In the fall of 1980 I taught my first class (AI) as a professor. The TA, Jeff Shrager, suggested this word problem, which he called dog-cat, as a good homework exercise for implementing the A* search algorithm. The students had to do it for three letter words in Lisp and on a Univac computer that the University of Pennsylvania’s Moore School had at the time. I’ve been using the problem on and off for 30 years now in homework assignments.
Saying that two groups are homomorphic doesn't tell me much more than there exists a group homomorphism... As far as I know, isomorphism implies one can be transformed into the other through renaming.
I don't know anything about topology, so I'll take your word for it.
http://research.microsoft.com/en-us/people/goldberg/