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.

Question: Is there a largest possible order of a finite group generated by $3$ class transpositions?

Remark: Given a bound $m \in \mathbb{N}$, let $n_{\rm max}(m)$ denote the largest possible order of a finite group generated by $3$ class transpositions interchanging residue classes with moduli $\leq m$. Then we have:

$n_{\rm max}(3) \ = \ 2^3 \cdot 3 \cdot 5$,

$n_{\rm max}(4) \ = \ 2^{15} \cdot 3^8 \cdot 5^3 \cdot 7^2 \cdot 11$,

$n_{\rm max}(5) \ = \ 2^{95} \cdot 3^{47} \cdot 5^{20} \cdot 7^{14} \cdot 11^7 \cdot 13^6 \cdot 17^3 \cdot 19^2 \cdot 23^2 \cdot 29^2 \cdot 31^2$,

$n_{\rm max}(6) \ = \ 2^{200} \cdot 3^{103} \cdot 5^{48} \cdot 7^{28} \cdot 11^{16} \cdot 13^{13} \cdot 17^8 \cdot 19^6 \cdot 23^6 \cdot 29$,

$n_{\rm max}(7) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$,

$n_{\rm max}(8) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$,

$n_{\rm max}(9) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$.

So maybe the sequence $(n_{\rm max}(m))$ gets constant already for $m \geq 7$, which would of course imply a positive answer to the question. In this case, $G = \langle \tau_{1(7),6(7)}, \tau_{0(5),3(5)}, \tau_{0(4),5(6)} \rangle$ would be a largest possible finite group generated by $3$ class transpositions.

Related questions:

• Transitivity on $\mathbb{N}_0$ — a 42 problem
• Groups generated by 3 involutions
• Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
• Characterization of the elements of an infinite simple group

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.

Question: Is there a largest possible order of a finite group generated by $3$ class transpositions?

Remark: Given a bound $m \in \mathbb{N}$, let $n_{\rm max}(m)$ denote the largest possible order of a finite group generated by $3$ class transpositions interchanging residue classes with moduli $\leq m$. Then we have:

$n_{\rm max}(3) \ = \ 2^3 \cdot 3 \cdot 5$,

$n_{\rm max}(4) \ = \ 2^{15} \cdot 3^8 \cdot 5^3 \cdot 7^2 \cdot 11$,

$n_{\rm max}(5) \ = \ 2^{95} \cdot 3^{47} \cdot 5^{20} \cdot 7^{14} \cdot 11^7 \cdot 13^6 \cdot 17^3 \cdot 19^2 \cdot 23^2 \cdot 29^2 \cdot 31^2$,

$n_{\rm max}(6) \ = \ 2^{200} \cdot 3^{103} \cdot 5^{48} \cdot 7^{28} \cdot 11^{16} \cdot 13^{13} \cdot 17^8 \cdot 19^6 \cdot 23^6 \cdot 29$,

$n_{\rm max}(7) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$,

$n_{\rm max}(8) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$,

$n_{\rm max}(9) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$.

So maybe the sequence $(n_{\rm max}(m))$ gets constant already for $m \geq 7$, which would of course imply a positive answer to the question. In this case, $G = \langle \tau_{1(7),6(7)}, \tau_{0(5),3(5)}, \tau_{0(4),5(6)} \rangle$ would be a largest possible finite group generated by $3$ class transpositions.

Related questions:

• Transitivity on $\mathbb{N}_0$ — a 42 problem
• Groups generated by 3 involutions
• Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
• Characterization of the elements of an infinite simple group

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.

Question: Is there a largest possible order of a finite group generated by $3$ class transpositions?

Remark: Given a bound $m \in \mathbb{N}$, let $n_{\rm max}(m)$ denote the largest possible order of a finite group generated by $3$ class transpositions interchanging residue classes with moduli $\leq m$. Then we have:

$n_{\rm max}(3) \ = \ 2^3 \cdot 3 \cdot 5$,

$n_{\rm max}(4) \ = \ 2^{15} \cdot 3^8 \cdot 5^3 \cdot 7^2 \cdot 11$,

$n_{\rm max}(5) \ = \ 2^{95} \cdot 3^{47} \cdot 5^{20} \cdot 7^{14} \cdot 11^7 \cdot 13^6 \cdot 17^3 \cdot 19^2 \cdot 23^2 \cdot 29^2 \cdot 31^2$,

$n_{\rm max}(6) \ = \ 2^{200} \cdot 3^{103} \cdot 5^{48} \cdot 7^{28} \cdot 11^{16} \cdot 13^{13} \cdot 17^8 \cdot 19^6 \cdot 23^6 \cdot 29$,

$n_{\rm max}(7) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$,

$n_{\rm max}(8) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$,

$n_{\rm max}(9) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$.

So maybe the sequence $(n_{\rm max}(m))$ gets constant already for $m \geq 7$, which would of course imply a positive answer to the question. In this case, $G = \langle (1(7),6(7)), (0(5),3(5)), (0(4),5(6)) \rangle$ would be a largest possible finite group generated by $3$ class transpositions.

Related questions:

• Transitivity on $\mathbb{N}_0$ — a 42 problem
• Groups generated by 3 involutions
• Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
• Characterization of the elements of an infinite simple group

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.

Question: Is there a largest possible order of a finite group generated by $3$ class transpositions?

Remark: Given a bound $m \in \mathbb{N}$, let $n_{\rm max}(m)$ denote the largest possible order of a finite group generated by $3$ class transpositions interchanging residue classes with moduli $\leq m$. Then we have:

$n_{\rm max}(3) \ = \ 2^3 \cdot 3 \cdot 5$,

$n_{\rm max}(4) \ = \ 2^{15} \cdot 3^8 \cdot 5^3 \cdot 7^2 \cdot 11$,

$n_{\rm max}(5) \ = \ 2^{95} \cdot 3^{47} \cdot 5^{20} \cdot 7^{14} \cdot 11^7 \cdot 13^6 \cdot 17^3 \cdot 19^2 \cdot 23^2 \cdot 29^2 \cdot 31^2$,

$n_{\rm max}(6) \ = \ 2^{200} \cdot 3^{103} \cdot 5^{48} \cdot 7^{28} \cdot 11^{16} \cdot 13^{13} \cdot 17^8 \cdot 19^6 \cdot 23^6 \cdot 29$,

$n_{\rm max}(7) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$,

$n_{\rm max}(8) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$,

$n_{\rm max}(9) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$.

So maybe the sequence $(n_{\rm max}(m))$ gets constant already for $m \geq 7$, which would of course imply a positive answer to the question. In this case, $G = \langle \tau_{1(7),6(7)}, \tau_{0(5),3(5)}, \tau_{0(4),5(6)} \rangle$ would be a largest possible finite group generated by $3$ class transpositions.

Related questions:

• Transitivity on $\mathbb{N}_0$ — a 42 problem
• Groups generated by 3 involutions
• Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
• Characterization of the elements of an infinite simple group

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.

Given a constant $C \in \mathbb{N}$, what is the largest possible order of a finite group generated by 3 class transpositions interchanging residue classes with moduli less than or equal to $C$?

The question asks for asymptotic behavior as well as for exact values for small $C$.

Remarks:

Groups generated by 3 class transpositions can either be finite or infinite. Among the 52394 unordered triples of pairwise distinct class transpositions interchanging residue classes with moduli less than or equal to $C = 6$, 21948 generate finite groups and 30446 generate infinite groups.

Largest finite groups for $C = 3, \dots, 6$ are as follows:

$C = 3$: $|<\tau_{(0(2),1(2)}, \tau_{0(3),1(3)}, \tau_{(1(3),2(3)}>| = 120 = 2^3 \cdot 3 \cdot 5$.

$C = 4$: $|<\tau_{(1(3),2(3)}, \tau_{(0(2),1(4)}, \tau_{2(4),3(4)}>| = 2^{15} \cdot 3^8 \cdot 5^3 \cdot 7^2 \cdot 11$.

$C = 5$: $|<\tau_{0(3),2(3)}, \tau_{1(2),0(4)}, \tau_{2(5),3(5)}>| =$ $2^{95} \cdot 3^{47} \cdot 5^{20} \cdot 7^{14} \cdot 11^7 \cdot 13^6 \cdot 17^3 \cdot 19^2 \cdot 23^2 \cdot 29^2 \cdot 31^2$.

$C = 6$: $|<\tau_{1(5),4(5)}, \tau_{0(3),1(6)}, \tau_{3(4),0(6)}>| =$ $2^{200} \cdot 3^{103} \cdot 5^{48} \cdot 7^{28} \cdot 11^{16} \cdot 13^{13} \cdot 17^8 \cdot 19^6 \cdot 23^6 \cdot 29$.

The mentioned groups and their orders have been determined with the GAP package RCWA, cf. http://www.gap-system.org/Packages/rcwa.html.

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.

Question: Is there a largest possible order of a finite group generated by $3$ class transpositions?

Remark: Given a bound $m \in \mathbb{N}$, let $n_{\rm max}(m)$ denote the largest possible order of a finite group generated by $3$ class transpositions interchanging residue classes with moduli $\leq m$. Then we have:

$n_{\rm max}(3) \ = \ 2^3 \cdot 3 \cdot 5$,

$n_{\rm max}(4) \ = \ 2^{15} \cdot 3^8 \cdot 5^3 \cdot 7^2 \cdot 11$,

$n_{\rm max}(5) \ = \ 2^{95} \cdot 3^{47} \cdot 5^{20} \cdot 7^{14} \cdot 11^7 \cdot 13^6 \cdot 17^3 \cdot 19^2 \cdot 23^2 \cdot 29^2 \cdot 31^2$,

$n_{\rm max}(6) \ = \ 2^{200} \cdot 3^{103} \cdot 5^{48} \cdot 7^{28} \cdot 11^{16} \cdot 13^{13} \cdot 17^8 \cdot 19^6 \cdot 23^6 \cdot 29$,

$n_{\rm max}(7) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$,

$n_{\rm max}(8) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$,

$n_{\rm max}(9) \ = \ 2^{1283} \cdot 3^{673} \cdot 5^{305} \cdot 7^{193} \cdot 11^{98} \cdot 13^{84} \cdot 17^{50} \cdot 19^{41} \cdot 23^{25} \cdot 29^{13} \cdot 31^4$.

So maybe the sequence $(n_{\rm max}(m))$ gets constant already for $m \geq 7$, which would of course imply a positive answer to the question. In this case, $G = \langle (1(7),6(7)), (0(5),3(5)), (0(4),5(6)) \rangle$ would be a largest possible finite group generated by $3$ class transpositions.

Related questions:

• Transitivity on $\mathbb{N}_0$ — a 42 problem
• Groups generated by 3 involutions
• Possible orders of products of 2 involutions which interchange disjoint residue classes of the integers
• Characterization of the elements of an infinite simple group