Problem Analysis #0

[THOUGHT] The group X is defined by the presentation $X = \langle u, v \mid u^{46} = 1, v^{40488236} = 1, vu = uv^{27636333} \rangle$. This presentation defines a semidirect product of the cyclic group $\langle u \rangle$ of order 46 and the cyclic group $\langle v \rangle$ of order 40488236. The relation $vu = uv^{27636333}$ can be rewritten as $vuv^{-1} = u^{27636333}$, which defines an automorphism of $\langle u \rangle$ given by multiplication by $k \equiv 27636333 \mod 46$. Computing $27636333 \mod 46$ yields 39, so the automorphism is multiplication by 39 modulo 46. The multiplicative order of 39 modulo 46 is 11, as $39^{11} \equiv 1 \pmod{46}$. Since the automorphism has finite order and the relations are satisfied, the group is finite. The order of the semidirect product is the product of the orders of the two cyclic groups, which is $46 \times 40488236$. Calculating this product gives $46 \times 40488236 = 1862458856$.

$\boxed{1862458856}$

[THOUGHT] We need to compute the order of the group X. First, compute k = 27636333 mod 46. Then check if k^40488236 ≡ 1 mod 46. If yes, the order is 46 * 40488236; else, 1. [/THOUGHT] [PYTHON] k = 27636333 % 46 m = 40488236 if pow(k, m, 46) == 1: order = 46 * m else: order = 1 print(order) [/PYTHON]

$\boxed{1862458856}$