This is a completely random question that just happened to come to mind recently and I was wondering if the MathSE community had anything to say about it.
Let $n > 1,b > 1$ be integers and consider the set $M_{n,b}$ of natural numbers whose base-$b$ representation has exactly $n$ digits including any leading zeros. Given a permutation $\varphi$ on $n$ letters and $m = m_1...m_n \in M_{n,b}$ define $\hat{\varphi}(m) = m_{\varphi(1)}...m_{\varphi(n)}$ so that $\hat{\varphi}$ permutes the digits of $m$ in accordance with $\varphi$.
For $k > 1$, define $S(n,b,k) = \{\varphi \in S_n: \hat{\varphi}(m) \!\!\!\!\! \mod k = m \!\!\!\!\! \mod k \text{ for all } m \in M_{n,b}\}$
Because $\varphi, \psi \in S(n,b,k)$ imply $\hat{\varphi \circ \psi}(m)\!\!\!\!\! \mod k = \hat{\varphi} \circ \hat{\psi}(m) \!\!\!\!\! \mod k$ $= \hat{\varphi}(m) \!\!\!\!\! \mod k = m \!\!\!\!\! \mod k$ and $S_n$ is finite, we see that $S(n,b,k)$ is a subgroup of $S_n$.
So my questions are:
- What are the properties of the groups $S(n,b,k)$?
- Are there any cases in which $S(n,b,k)$ is isomorphic to a known group?
- Have the groups $S(n,b,k)$ been studied before? If so, what are known properties?
I realize that these questions are very broad, but any interesting information that comes to your head will suffice as an answer.
There are some special cases in which the question can be answered quickly. For example, $S(n, 10, 3) = S_n \text{ for all } n > 0$ This can be seen by remembering that the remainder of $n$ divided by $3$ is the same as the remainder of the sum of the digits of $n$ divided by $3$ and noticing that permuting the digits does not change their sum. This can be generalized to the cases where $k$ divides $b-1$ and $n$ is arbitrary. I also imagine that a similar analysis will yield results for the case in which $k$ divides $b+1$ by recalling the test for divisibility by $11$.
Any thoughts?