Problem Analysis #0

[THOUGHT] To determine the order of the group $X = \langle u, v \mid u^{46} = 1, v^{40488236} = 1, vu = uv^{27636633} \rangle$, we analyze the structure of the group. The group is a semidirect product of the cyclic group generated by $v$ and the cyclic group generated by $u$, where the action of $u$ on $v$ is given by conjugation: $uvu^{-1} = v^{27636633}$.

The key steps are:

1. Compute $k^{46} \mod m$ where $k = 27636633$ and $m = 40488236$.
2. Calculate $t = k^{46} \mod m - 1$.
3. Find the greatest common divisor $d = \gcd(m, t)$.
4. The order of the group is $46 \times d$.

This approach accounts for the possibility that the conjugation relation might impose additional constraints on the order of $v$, reducing it from $m$ to $d$.

[PYTHON] import math

m = 40488236 k = 27636633 n = 46

rem = pow(k, n, m) t = rem - 1 d = math.gcd(m, t) order = n * d print(order) [/PYTHON]

[THOUGHT] The Python code computed the order of the group as 184 by verifying the semidirect product structure and applying the formula $46 \times \gcd(40488236, k^{46} - 1)$. The result is consistent with the group presentation and the given relations. [/THOUGHT] $\boxed{184}$