From an applied maths background, I'm familiar with (binary) bit-reversal involutions and more generally, radix-reversal involutions when using a mixed-radix counting system.
I've become interested in these from a pure maths viewpoint, but haven't been able to find much on their group theory. Even simple questions such as 'what subgroups of symmetric groups do they generate'?
There's plenty of literature on them in applied maths & computer science due to their use in Fast Fourier Transforms, but I'm looking for pointers to their pure group theory. I'm sure this must have been well-studied, so any decent references will be well-appreciated :)
Edited to add, thanks for the replies! To be more precise about the 'subgroups of symmetric groups' question, this is what I'm looking at :
Given some non-prime natural number N, along with an ordered factorisation N = a_1a_2..a_k (not necessarily a prime factorisation), we can write each natural number between 0 and N-1 using the a_1 a_2 .. a_k radix counting system, and derive an involution by radix reversal. Given a different ordered factorisation, we derive a different involution on the same set of naturals. I'm interesting in characterising the subgroup of the symmetric group on {0,..,N-1} that these involutions generate.