Abstract Algebra: Theory and Applications · Problem 15.8.3
Rings — Thomas Judson
Mathematics — Abstract Algebra · sylow-theorems
Problem
For the dihedral group D36 (symmetries of a 36-gon) and each possible prime, determine the possibilities for the number of distinct Sylow p-subgroups as predicted by the Third Sylow Theorem (Theorem 15.1.8). Now compute the actual number of distinct Sylow p-subgroups for each prime and comment on the result.
It can be proved that any group with order 72 is not a simple group, using techniques such as those used in the later examples in this chapter. Discuss this result in the context of your computations with Sage.