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Group Actions and Symmetry

This quiz covers fundamental concepts of group actions, including orbits, stabilizers, and the orbit-stabilizer theorem, as well as important symmetry groups such as permutation groups, dihedral groups, and polyhedral groups.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is a group action?

  1. A mapping from a group to a set
  2. A transformation of a set by a group
  3. A group of transformations of a set
  4. A set of transformations that preserve a given structure
Question 2 Multiple Choice (Single Answer)

What is the orbit of an element under a group action?

  1. The set of all elements that are transformed into the given element
  2. The set of all elements that transform the given element into itself
  3. The set of all elements that commute with the given element
  4. The set of all elements that are fixed by the given element
Question 3 Multiple Choice (Single Answer)

What is the stabilizer of an element under a group action?

  1. The set of all elements that are transformed into the given element
  2. The set of all elements that transform the given element into itself
  3. The set of all elements that commute with the given element
  4. The set of all elements that are fixed by the given element
Question 4 Multiple Choice (Single Answer)

What is the kernel of a group action?

  1. The set of all elements that are transformed into the identity element
  2. The set of all elements that transform the identity element into itself
  3. The set of all elements that commute with the identity element
  4. The set of all elements that are fixed by the identity element
Question 5 Multiple Choice (Single Answer)

What is the fixed point set of a group action?

  1. The set of all elements that are transformed into themselves
  2. The set of all elements that transform themselves into themselves
  3. The set of all elements that commute with themselves
  4. The set of all elements that are fixed by themselves
Question 6 Multiple Choice (Single Answer)

What is the isotropy subgroup of an element under a group action?

  1. The set of all elements that are transformed into the given element
  2. The set of all elements that transform the given element into itself
  3. The set of all elements that commute with the given element
  4. The set of all elements that are fixed by the given element
Question 7 Multiple Choice (Single Answer)

What is the orbit-stabilizer theorem?

  1. The order of the orbit of an element is equal to the index of the stabilizer of the element
  2. The order of the stabilizer of an element is equal to the index of the orbit of the element
  3. The order of the orbit of an element is equal to the order of the stabilizer of the element
  4. The order of the stabilizer of an element is equal to the order of the orbit of the element
Question 8 Multiple Choice (Single Answer)

What is a transitive group action?

  1. A group action in which every element of the set is transformed into every other element
  2. A group action in which every element of the set is transformed into itself
  3. A group action in which every element of the set is transformed into a unique element
  4. A group action in which every element of the set is transformed into a finite number of elements
Question 9 Multiple Choice (Single Answer)

What is a regular group action?

  1. A transitive group action in which the stabilizer of every element is trivial
  2. A transitive group action in which the stabilizer of every element is finite
  3. A transitive group action in which the stabilizer of every element is infinite
  4. A transitive group action in which the stabilizer of every element is non-trivial
Question 10 Multiple Choice (Single Answer)

What is a permutation group?

  1. A group of transformations of a set
  2. A group of transformations of a set that preserves a given structure
  3. A group of transformations of a set that is transitive
  4. A group of transformations of a set that is regular
Question 11 Multiple Choice (Single Answer)

What is a symmetric group?

  1. A permutation group that is transitive
  2. A permutation group that is regular
  3. A permutation group that is both transitive and regular
  4. A permutation group that is neither transitive nor regular
Question 12 Multiple Choice (Single Answer)

What is the alternating group?

  1. The subgroup of the symmetric group consisting of all even permutations
  2. The subgroup of the symmetric group consisting of all odd permutations
  3. The subgroup of the symmetric group consisting of all permutations that fix a given element
  4. The subgroup of the symmetric group consisting of all permutations that do not fix any element
Question 13 Multiple Choice (Single Answer)

What is the dihedral group?

  1. The group of symmetries of a regular polygon
  2. The group of symmetries of a regular polyhedron
  3. The group of symmetries of a sphere
  4. The group of symmetries of a cube
Question 14 Multiple Choice (Single Answer)

What is the octahedral group?

  1. The group of symmetries of a cube
  2. The group of symmetries of a regular octahedron
  3. The group of symmetries of a regular dodecahedron
  4. The group of symmetries of a regular icosahedron
Question 15 Multiple Choice (Single Answer)

What is the icosahedral group?

  1. The group of symmetries of a cube
  2. The group of symmetries of a regular octahedron
  3. The group of symmetries of a regular dodecahedron
  4. The group of symmetries of a regular icosahedron
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