Generalized Quaternions

The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomorphism). The other one, D4, can be constructed as a semi-direct product: D4 ∼= Aff(Z/(4)) ∼= Z/(4) o (Z/(4))× ∼= Z/(4) o Z/(2), where the elements of Z/(2) act on Z/(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We will see how this is done in Section 2 and then jazz up the construction in Section 3 to make an infinite family of similar groups with Q8 as the simplest member. In Section 4 we will compare this family with the dihedral groups and see how it fits into a bigger picture.