## Definition / Question ##

<b> Definition:</b> Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where
$0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the *class transposition*
$\tau_{r_1(m_1),r_2(m_2)}$ be the permutation of $\mathbb{Z}$ which interchanges
$r_1+km_1$ and $r_2+km_2$ for every $k \in \mathbb{Z}$ and which fixes everything else.

> <b>Question:</b> Which integers occur as orders of products of 2 class transpositions?

## Known results ##

The known finite orders of products of 2 class transpositions are the divisors of 60
except for 5, as well as 8, 24, 40, 42, 84, 120, 168 and 420. I do not know whether
there are further, or even only whether there are finitely or infinitely many.

Among the 409965 unordered pairs of distinct class transpositions which interchange
residue classes with moduli $\leq 12$, there are

- 179470 whose product has order 6,

- 83298 whose product has order 12,

- 60208 whose product has order $\infty$,

- 38818 whose product has order 2,

- 14127 whose product has order 3,

- 13491 whose product has order 4,

- 10407 whose product has order 60,

- 8918 whose product has order 30,

- 976 whose product has order 20,

- 218 whose product has order 10,

- 32 whose product has order 15, and

- 2 whose product has order 8.

The orders 24, 40, 42, 84, 120, 168 and 420 appear to be still more "seldom",
and occur only for products of class transpositions which interchange residue
classes with larger moduli.

## Remarks / Background ##

1. I asked this question on the [Nikolaus Conference 2012][1] in Aachen.
   In response, [Michael Cuntz][2] found some possible orders which were not known
   to me at that time (namely 8, 24 and 42).
   Afterwards, a more systematic search by myself revealed the further possible
   orders 40, 84, 120, 168 and 420. The question though remains open.

2. Some products of 2 class transpositions have infinite order, but the length
   of the longest cycle which intersects nontrivially with a set $\{1, \dots, n\}$
   grows only logarithmically with $n$. This may fool the naive approach of
   estimating the order by taking a number of cycles and computing the
   least common multiple of their lengths.

3. The set of all class transpositions of $\mathbb{Z}$ generates the infinite
   simple group discussed in the article

   [A Simple Group Generated by Involutions Interchanging Residue Classes of the Integers][3].
   *Math. Z.* 264 (2010), no. 4, 927-938.

## Examples ##

Minimal examples for the known orders are as follows ("minimal" means in
this context that the maximum of the moduli of the involved residue classes
is the smallest possible):

- Order $\infty$: $\tau_{0(2),1(4)} \cdot \tau_{0(2),1(2)}$

- Order 2: $\tau_{0(4),1(4)} \cdot \tau_{2(4),3(4)}$

- Order 3: $\tau_{0(3),1(3)} \cdot \tau_{0(3),2(3)}$

- Order 4: $\tau_{0(3),1(3)} \cdot \tau_{0(2),1(2)}$

- Order 6: $\tau_{0(3),2(3)} \cdot \tau_{0(2),1(2)}$

- Order 8: $\tau_{0(2),5(6)} \cdot \tau_{2(9),8(9)}$

- Order 10: $\tau_{1(2),2(4)} \cdot \tau_{0(3),4(6)}$

- Order 12: $\tau_{1(4),3(4)} \cdot \tau_{1(3),2(3)}$

- Order 15: $\tau_{0(3),2(3)} \cdot \tau_{0(2),1(4)}$

- Order 20: $\tau_{1(4),3(4)} \cdot \tau_{0(3),1(6)}$

- Order 24: $\tau_{0(3),1(6)} \cdot \tau_{1(8),19(20)}$

- Order 30: $\tau_{0(3),1(3)} \cdot \tau_{0(2),3(4)}$

- Order 40: $\tau_{0(10),3(15)} \cdot \tau_{2(4),12(24)}$

- Order 42: $\tau_{0(4),9(10)} \cdot \tau_{0(3),4(15)}$

- Order 60: $\tau_{3(4),4(6)} \cdot \tau_{1(5),2(5)}$

- Order 84: $\tau_{0(8),6(20)} \cdot \tau_{0(3),26(30)}$

- Order 120: $\tau_{0(4),10(24)} \cdot \tau_{1(15),10(30)}$

- Order 168: $\tau_{0(8),6(20)} \cdot \tau_{2(3),6(30)}$

- Order 420: $\tau_{0(8),6(20)} \cdot \tau_{6(15),26(30)}$

  [1]: http://www.math.rwth-aachen.de:8001/Nikolaus2012/index.html
  [2]: http://www.mathematik.uni-kl.de/~cuntz/de/index.html
  [3]: http://dx.doi.org/10.1007/s00209-009-0497-8