Generalized quaternion group Q_32

the generalized quaternion group $Q_{32}$ is the group with presentation

Q_{32}=\langle a,b\mid a^{16}=1,\quad b^2=a^{8},\quad bab^{-1}=a^{-1}\rangle.

Order and nonsimplicity

Every element has a unique form $a^j$ or $a^jb$ , with $0\leq j<16$ , so the order is $32$ . The element $a^{8}$ has order two and generates the center. This proper nontrivial normal subgroup shows that the group is not simple.

Smallest member

At $n=3$ , the assignments $a\mapsto i$ , $b\mapsto j$ identify this presentation with the quaternion group $Q_8$ .

References

J. S. Milne, Group Theory, v4.01, §1.18, p. 14; Example 2.7(b), p. 35: generalized quaternion groups.