Exercise · ZW-BURNSIDE-S-LEMMA-010 · ★★★ Advanced
A group of 6 elements acts on a set X. The sum of the fixed points of all its elements, \(\sum_{g}|X^g|\), is 20. What can be concluded?
Choose one option.
A There are 6 orbits B There are 3 orbits C There are 20 orbits D The data are inconsistent, because \(20/6\) is not an integer Show answer D. The data are inconsistent, because \(20/6\) is not an integer← Back to the question
Explanation Burnside gives the number of orbits as \(\sum_g |X^g|/|G|\); \(20/6\) is not an integer, so these data cannot come from a real group action .
Burnside's lemma (also called the Cauchy-Frobenius lemma ) states that the number of orbits of a finite group G acting on a set X equals the average number of fixed points of the elements of G.
\[|X/G| = \frac{1}{|G|}\sum_{g\in G}|X^g|\]
More practice problems: burnside's lemma