Abstract Algebra: Theory and Applications · Problem 14.5.3
The Sylow Theorems — Thomas Judson
Mathematics — Abstract Algebra · g-equivalence-classes-orbits
Problem
Compute the -equivalence classes of for each of the -sets in Exercise 14.5.2. For each verify that . Hint. (a) .Intuitive explanation
The orbit-stabilizer theorem is the key insight here: it tells you that the size of a group always factors as the product of an orbit's size and the stabilizer's size. To find equivalence classes, identify which elements can be reached from each other under the group action—these form orbits. Then for each element, count how many group elements fix it (the stabilizer). The theorem guarantees these two counts will always multiply to give , providing a powerful consistency check on your work.
Intuitive explanation
The orbit-stabilizer theorem is the key insight here: it tells you that the size of a group always factors as the product of an orbit's size and the stabilizer's size. To find equivalence classes, identify which elements can be reached from each other under the group action—these form orbits. Then for each element, count how many group elements fix it (the stabilizer). The theorem guarantees these two counts will always multiply to give , providing a powerful consistency check on your work.