Finite Subgroups of the Quaternions

10/28/2014 - 12:15pm
10/28/2014 - 1:10pm
Will Murray (California State University Long Beach)

Question:  Given a field, what interesting finite multiplicative groups does it contain? 

Answer:  None.  In algebra class we prove that any finite group in a field is cyclic. 

However, division rings (noncommutative fields) are much more interesting.  At the very least, we know that the quaternions contain the finite quaternion group {1,-1,i,-i,j,-j,k,-k}, which is not cyclic.  Lots of interesting geometry gives us a classification of all finite subgroups of the quaternions.  They correspond to subgroups of SU(2) and SO(3), which correspond in turn to rotation groups of the Platonic solids. If you know basic group theory, you'll be able to understand the talk.  Knowing the definitions of SU(2) and SO(3) would help, but we'll review those as we go along.

MDSL 126