Generalized quaternion group Q_(2^n)

For an integer $n\geq3$ , the generalized quaternion group $Q_{2^n}$ is the group with presentation

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

Order and nonsimplicity

Every element has a unique form $a^j$ or $a^jb$ , with $0\leq j<2^{n-1}$ , so the order is $2^n$ . The element $a^{2^{n-2}}$ 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.