Quaternion Group - Generalized Quaternion Group

A group is called a generalized quaternion group or dicyclic group if it has a presentation

for some integer n ≥ 2. This group is denoted Q_4n and has order 4n. Coxeter labels these dicyclic groups <2,2,n>, being a special case of the binary polyhedral group and related to the polyhedral groups (p,q,r), and dihedral group (2,2,n). The usual quaternion group corresponds to the case n = 2. The generalized quaternion group can be realized as the subgroup of GL₂(C) generated by

where ω_n = eiπ/n. It can also be realized as the subgroup of unit quaternions generated by x = eiπ/n and y = j.

The generalized quaternion groups have the property that every abelian subgroup is cyclic. It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above. Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or generalized quaternion (of order a power of 2). In particular, for a finite field F with odd characteristic, the 2-Sylow subgroup of SL₂(F) is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, (Gorenstein 1980, p. 42). Letting pr be the size of F, where p is prime, the size of the 2-Sylow subgroup of SL₂(F) is 2n, where n = ord₂(p2 - 1) + ord₂(r).

The Brauer-Suzuki theorem shows that groups whose Sylow 2-subgroup is generalized quaternion cannot be simple.