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Group Theory

Group Theory Quiz

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is the identity element of a group?

  1. The element that is its own inverse
  2. The element that is the same as every other element
  3. The element that is the only element of the group
  4. The element that is the first element of the group
Question 2 Multiple Choice (Single Answer)

What is the order of an element in a group?

  1. The number of elements in the group
  2. The number of times the element appears in the group
  3. The smallest positive integer $n$ such that $a^n = e$
  4. The largest positive integer $n$ such that $a^n = e$
Question 3 Multiple Choice (Single Answer)

What is a subgroup of a group?

  1. A set of elements that is closed under the group operation
  2. A set of elements that contains the identity element
  3. A set of elements that is non-empty
  4. A set of elements that is finite
Question 4 Multiple Choice (Single Answer)

What is a normal subgroup of a group?

  1. A subgroup that is invariant under conjugation
  2. A subgroup that is closed under the group operation
  3. A subgroup that contains the identity element
  4. A subgroup that is non-empty
Question 5 Multiple Choice (Single Answer)

What is the quotient group of a group by a normal subgroup?

  1. The group obtained by dividing the group by the normal subgroup
  2. The group obtained by taking the union of the group and the normal subgroup
  3. The group obtained by taking the intersection of the group and the normal subgroup
  4. The group obtained by taking the complement of the normal subgroup in the group
Question 6 Multiple Choice (Single Answer)

What is the center of a group?

  1. The set of all elements that commute with every other element in the group
  2. The set of all elements that are their own inverses
  3. The set of all elements that are in the center of the group
  4. The set of all elements that are in the center of the group
Question 7 Multiple Choice (Single Answer)

What is the commutator subgroup of a group?

  1. The subgroup generated by all the commutators of the group
  2. The subgroup generated by all the elements of the group
  3. The subgroup generated by all the inverses of the elements of the group
  4. The subgroup generated by all the squares of the elements of the group
Question 8 Multiple Choice (Single Answer)

What is the derived subgroup of a group?

  1. The subgroup generated by all the commutators of the group
  2. The subgroup generated by all the elements of the group
  3. The subgroup generated by all the inverses of the elements of the group
  4. The subgroup generated by all the squares of the elements of the group
Question 9 Multiple Choice (Single Answer)

What is the free group on a set of generators?

  1. The group generated by the set of generators subject to no relations
  2. The group generated by the set of generators subject to the relation that every generator is its own inverse
  3. The group generated by the set of generators subject to the relation that every generator commutes with every other generator
  4. The group generated by the set of generators subject to the relation that every generator is equal to the identity element
Question 10 Multiple Choice (Single Answer)

What is the fundamental group of a space?

  1. The group of all homotopy classes of loops in the space
  2. The group of all homotopy classes of paths in the space
  3. The group of all homotopy classes of spheres in the space
  4. The group of all homotopy classes of disks in the space
Question 11 Multiple Choice (Single Answer)

What is the homology group of a space?

  1. The group of all homology classes of cycles in the space
  2. The group of all homology classes of boundaries in the space
  3. The group of all homology classes of spheres in the space
  4. The group of all homology classes of disks in the space
Question 12 Multiple Choice (Single Answer)

What is the cohomology group of a space?

  1. The group of all cohomology classes of cocycles in the space
  2. The group of all cohomology classes of coboundaries in the space
  3. The group of all cohomology classes of spheres in the space
  4. The group of all cohomology classes of disks in the space
Question 13 Multiple Choice (Single Answer)

What is the K-theory group of a space?

  1. The group of all K-theory classes of vector bundles in the space
  2. The group of all K-theory classes of sphere bundles in the space
  3. The group of all K-theory classes of disk bundles in the space
  4. The group of all K-theory classes of line bundles in the space
Question 14 Multiple Choice (Single Answer)

What is the L-theory group of a space?

  1. The group of all L-theory classes of generalized homology theories in the space
  2. The group of all L-theory classes of cohomology theories in the space
  3. The group of all L-theory classes of K-theory theories in the space
  4. The group of all L-theory classes of stable homotopy theories in the space
Question 15 Multiple Choice (Single Answer)

What is the motivic cohomology group of a space?

  1. The group of all motivic cohomology classes of algebraic varieties in the space
  2. The group of all motivic cohomology classes of schemes in the space
  3. The group of all motivic cohomology classes of stacks in the space
  4. The group of all motivic cohomology classes of categories in the space
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