Is this semigroup a group?

Question

I have the following structure $$\langle a,b\mid a^5=a, b^9=b, a^2=b^2, ab=b^7a\rangle$$ GAP tells me that this is of order $8$. Is it probably the quaternion group? How could GAP show it? Thanks!

Answer 1

The quaternion group with the usual notation satisfies these equalities: \begin{align} ij & = k & ji & = -k \\ jk & = i & kj & = -i \\ ki & = j & ik & = -j \\ i^2 = j^2 = k^2 & = -1 \end{align} What you have is consistent with $a=i$ and $b=j$.

It is not consistent with the dihedral group of order $8$. If two elements $a,b$ generate the dihedral group, then at least one of them must have order $2$, since all the other elements only generate the rotation group of order $4$. But they cannot both have order $2$, since then they generate a subgroup of order $4$ in which each non-unit element has order $2$. So if they're in the dihedral group, we'll have to assume one has order $4$ and the other has order $2$. But that is not consistent with $a^2=b^2$.

I'll leave it as an exercise to rule out the three abelian groups of order $8$.

Hence if this is a non-abelian group of order $8$, then it's the quaternion group.