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1. I asked this question on the Nikolaus Conference 2012 in Aachen. In response, Michael Cuntz 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. Math. Z. 264 (2010), no. 4, 927-938. (PDF).

• a database of pairs of class transpositions which interchange residue classes with moduli $\leq 32$ and whose product has finite order not dividing 60, i.e. the "seldom" ones -- see here (162KB), and

• a database of all 409965 pairs of class transpositions which interchange residue classes with moduli $\leq 12$, sorted by the order of their product -- see here (18MB).

1. I asked this question on the Nikolaus Conference 2012 in Aachen. In response, Michael Cuntz 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. Math. Z. 264 (2010), no. 4, 927-938. (PDF).

• a database of pairs of class transpositions which interchange residue classes with moduli $\leq 32$ and whose product has finite order not dividing 60, i.e. the "seldom" ones -- see here (162KB), and

• a database of all 409965 pairs of class transpositions which interchange residue classes with moduli $\leq 12$, sorted by the order of their product -- see here (18MB).

1. I asked this question on the Nikolaus Conference 2012 in Aachen. In response, Michael Cuntz 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. Math. Z. 264 (2010), no. 4, 927-938.

1. I asked this question on the Nikolaus Conference 2012 in Aachen. In response, Michael Cuntz 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. Math. Z. 264 (2010), no. 4, 927-938. (PDF).

The GAP package RCWA contains

• a database of pairs of class transpositions which interchange residue classes with moduli $\leq 32$ and whose product has finite order not dividing 60, i.e. the "seldom" ones -- see here (162KB), and

• a database of all 409965 pairs of class transpositions which interchange residue classes with moduli $\leq 12$, sorted by the order of their product -- see here (18MB).