The word problem for finite groups is decidable. Is it obvious that this is true?
In particular, I'm not entirely sure about what it means for the problem to be decidable (in this case---I think I understand what decidable means in general). I assume it means that we are given a fixed group G (do we have to assume this), but is the generating set (the letters) also fixed?
To decide the word problem for the group of symmetries of a square, with reflection $r$ and transposition $t$ as the generators, I would first find a canonical form for the group elements $1,r,r^2,r^3,t,tr,tr^2,tr^3$, and then note that using the relations $r^4=1$ and $rt=tr^3$ to show that any word can be reduced to something on my list. However, this seems like a lot of work, and it's not obvious to me what should be done in the case of an arbitrary finite group.