Problem Analysis #0

[THOUGHT] We have a group presentation $X=\langle u,v\mid u^{46}=1,\ v^{40{,}488{,}236}=1,\ vu=uv^{27{,}636{,}633}\rangle$. Rewrite the relation as $v^{-1}uv=u v^{27{,}636{,}633-1}=u v^{27{,}636{,}632}$. In a two-generator group with one conjugation relation, $u$ normalizes $\langle v\rangle$, so $\langle u,v\rangle$ is a semidirect product $\langle u\rangle \ltimes \langle v\rangle$. Thus

provided the conjugation does not force additional collapsing (which it does not here because the exponents are positive and $\langle v\rangle$ is cyclic).

So $|X|=46\cdot 40{,}488{,}236$. Compute this exactly in Python. [/THOUGHT] [PYTHON] print(46 * 40_488_236) [/PYTHON]

[THOUGHT] The group $X$ is generated by $u$ and $v$ with orders $46$ and $40{,}488{,}236$, and the relation $vu=uv^{27{,}636{,}633}$ does not reduce these orders, so $|X|=46\cdot 40{,}488{,}236=1862458856$. [/THOUGHT] \boxed{1862458856}