Problem Analysis #0

The group $X$ is defined by the presentation:

From the relation $vu = uv^{27636633}$, it follows that $u^{-1} v u = v^{27636633}$. Conjugating $v$ by $u$ gives:

Setting $i = 46$ and using $u^{46} = 1$:

so:

The order of $v$ must divide both $40488236$ and $(27636633)^{46} - 1$. Computing the greatest common divisor:

Thus, the order of $v$ divides 4. Additionally, the relation $vu = uv^{27636633}$ and the fact that $27636633 \equiv 1 \pmod{4}$ imply that $v$ commutes with $u$, so $[v, u] = 1$. This commutator relation, combined with the other relations, leads to $v^{27636632} = 1$ (since $k - 1 = 27636632$), which is consistent with the order of $v$ dividing 4.

Given that the order of $v$ divides 4 and no relations force a smaller order, the order of $v$ is exactly 4. The order of $u$ is 46, and since $u$ and $v$ commute and their orders are coprime in the sense that the group is abelian with no additional identifications, $X$ is isomorphic to $\mathbb{Z}/46\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$. The order of this group is:

All relations are satisfied:

• $u^{46} = 1$,
• $v^{40488236} = (v^4)^{10122059} = 1^{10122059} = 1$ (since $4 \mid 40488236$),
• $vu = uv$ (since $v$ and $u$ commute), and $uv^{27636633} = uv^{27636632 + 1} = u v^{k-1} v = u \cdot 1 \cdot v = uv$, so $vu = uv^{27636633}$.

Thus, the order of $X$ is 184.

$\boxed{184}$