Finding groups from presentations

Question

I am trying to solve the following problem: Find all groups with two generators $a$ and $b$ in which $a^4 = 1, b^2 = a^2, $ and $bab^{-1} = a^{-1}.$

I know that this is a presentation of the quaternion group, and am confused about whether this presentation can be a part of the presentation of some other group. Is it possible that this can be part of the presentation of another group? How would I construct another group?

Answer 1

Let $Q_8 = \langle\, a, b \mid a^4 = 1,\ b^2 = a^2,\ bab^{-1} = a^{-1} \,\rangle$. For every group $G$ with generators satisfying the relations, there is a surjection from $Q_8$ to $G$ by the von Dyck theorem or by the universal property of group presentation. (See [Aschbacher 2000, (28.6)] or [Grillet 2007, Theorem 7.2].) Thus $G$ is isomorphic to one of

• trivial group — $1$,
• group of order $2$ — $C_2$,
• Klein four group — $V_4$, or
• the whole group — $Q_8$.