I recently answered the question on How to find the order of a Rubik's cube algorithm. As you can see in my answer to Creating a Rubik's Cube Algorithm, I mentioned that using repetition is one of several ways to create algorithms to solve the cube.
Are there any examples of twisty permutation puzzles which there is no other logical way to find algorithms for (by hand, not with a computer search) except by, perhaps, having knowledge of how to compute the cyclic order of a permutation?
As those who have been following my work (and my answers to Rubik's Cube-related questions on math.stackexchange) may be aware, I have quite a history with finding what is called "$4\times4\times4$ parity algorithms".
Despite all of the experience I have with finding $4\times4\times4$ parity algorithms by hand, I would not have been able to find a parity algorithm whose types of moves are limited by a particular restriction, should I have not known about the order of a permutation.
About a year ago, I published this video series on 2-gen $4\times4\times4$ parity algorithm move sequences. Basically, there are five different 2-gen movesets on the $4\times4\times4$ Rubik's Cube which we can (with sufficient moves) create a "single dedge flip" with. In the final video of the series, I illustrate how we can apply the logic of calculating the order of an algorithm to be able to find a "single dedge flip" algorithm in the very restrictive move set it addresses---this would be a very difficult (and implausible) task otherwise.
A few final details on this particular feat:
I think that this idea may apply to other extraordinarily difficult puzzles or puzzles in which we restrict which moves can be turned (like bandaged twisty permutation puzzles).
In conclusion, this is also one instance in which group theory was directly used to solve a permutation twisty puzzle.
