Home › Quizzes › Burnside's Lemma

Burnside's Lemma

This quiz is designed to test your understanding of Burnside's Lemma, a fundamental result in combinatorics that relates the number of orbits of a group action to the number of fixed points.

15 Questions Published
Play Quiz Practise this topic out loud

Questions

Question 1 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. The number of orbits of $G$ on $X$ is equal to:

  1. The number of elements in $G$
  2. The number of elements in $X$
  3. The number of fixed points of $G$ on $X$
  4. The average number of elements in each orbit
Question 2 Multiple Choice (Single Answer)

If a group $G$ acts on a set $X$ and every element of $X$ is fixed by every element of $G$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 3 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $H$ is a subgroup of $G$, then the number of orbits of $H$ on $X$ is:

  1. Less than or equal to the number of orbits of $G$ on $X$
  2. Equal to the number of orbits of $G$ on $X$
  3. Greater than or equal to the number of orbits of $G$ on $X$
  4. Unrelated to the number of orbits of $G$ on $X$
Question 4 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is transitive on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 5 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is regular on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 6 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is doubly transitive on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 7 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is triply transitive on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 8 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is $k$-transitive on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 9 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is primitive on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 10 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is imprimitive on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 11 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is sharply transitive on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 12 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is semiregular on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 13 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is quasiprimitive on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 14 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is doubly quasiprimitive on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
Question 15 Multiple Choice (Single Answer)

Let $G$ be a group acting on a set $X$. If $G$ is triply quasiprimitive on $X$, then the number of orbits of $G$ on $X$ is:

  1. 0
  2. 1
  3. $|G|$
  4. $|X|$
← Back to Quizzes