Abstract Algebra: Theory and Applications · Problem 15.8.2
Rings — Thomas Judson
Mathematics — Abstract Algebra · sylow-theorems
Problem
For each possible prime, find all of the distinct Sylow p-subgroups of the alternating group A5. Confirm that your results are consistent with the Third Sylow Theorem for each prime. We know that A5 is a simple group. Explain how this would explain or predict some aspects of your answers.
Count the number of distinct elements contained in the union of all the Sylow subgroups you just found. What is interesting about this count?