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Representation Theory

Representation Theory Quiz

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is a representation of a group?

  1. A homomorphism from the group to the group of invertible linear transformations of a vector space.
  2. A homomorphism from the group to the group of invertible linear transformations of a module.
  3. A homomorphism from the group to the group of invertible linear transformations of a ring.
  4. A homomorphism from the group to the group of invertible linear transformations of a field.
Question 2 Multiple Choice (Single Answer)

What is the dimension of a representation?

  1. The number of elements in the group.
  2. The number of generators of the group.
  3. The number of conjugacy classes in the group.
  4. The number of irreducible representations of the group.
Question 3 Multiple Choice (Single Answer)

What is the character of a representation?

  1. The trace of the representation.
  2. The determinant of the representation.
  3. The eigenvalues of the representation.
  4. The eigenvectors of the representation.
Question 4 Multiple Choice (Single Answer)

What is the Schur orthogonality relation?

  1. The inner product of two characters of a representation is zero if the representations are not equivalent.
  2. The inner product of two characters of a representation is one if the representations are equivalent.
  3. The inner product of two characters of a representation is equal to the number of conjugacy classes in the group.
  4. The inner product of two characters of a representation is equal to the order of the group.
Question 5 Multiple Choice (Single Answer)

What is the Maschke theorem?

  1. Every representation of a group is completely reducible.
  2. Every representation of a group is semisimple.
  3. Every representation of a group is irreducible.
  4. Every representation of a group is trivial.
Question 6 Multiple Choice (Single Answer)

What is the Wedderburn theorem?

  1. Every semisimple ring is isomorphic to a direct sum of matrix rings.
  2. Every semisimple ring is isomorphic to a direct sum of division rings.
  3. Every semisimple ring is isomorphic to a direct sum of fields.
  4. Every semisimple ring is isomorphic to a direct sum of modules.
Question 7 Multiple Choice (Single Answer)

What is the Artin-Wedderburn theorem?

  1. Every semisimple algebra is isomorphic to a direct sum of matrix algebras.
  2. Every semisimple algebra is isomorphic to a direct sum of division algebras.
  3. Every semisimple algebra is isomorphic to a direct sum of fields.
  4. Every semisimple algebra is isomorphic to a direct sum of modules.
Question 8 Multiple Choice (Single Answer)

What is the Jacobson radical of a ring?

  1. The largest nilpotent ideal of the ring.
  2. The largest radical ideal of the ring.
  3. The largest prime ideal of the ring.
  4. The largest maximal ideal of the ring.
Question 9 Multiple Choice (Single Answer)

What is the Noether-Skolem theorem?

  1. Every semisimple ring is a direct sum of simple rings.
  2. Every semisimple ring is a direct sum of division rings.
  3. Every semisimple ring is a direct sum of fields.
  4. Every semisimple ring is a direct sum of modules.
Question 10 Multiple Choice (Single Answer)

What is the Krull-Schmidt theorem?

  1. Every semisimple module is a direct sum of simple modules.
  2. Every semisimple module is a direct sum of indecomposable modules.
  3. Every semisimple module is a direct sum of projective modules.
  4. Every semisimple module is a direct sum of injective modules.
Question 11 Multiple Choice (Single Answer)

What is the Brauer-Thrall theorem?

  1. Every semisimple algebra is a direct sum of simple algebras.
  2. Every semisimple algebra is a direct sum of division algebras.
  3. Every semisimple algebra is a direct sum of fields.
  4. Every semisimple algebra is a direct sum of modules.
Question 12 Multiple Choice (Single Answer)

What is the Frobenius-Schur theorem?

  1. Every semisimple group is a direct product of simple groups.
  2. Every semisimple group is a direct product of division groups.
  3. Every semisimple group is a direct product of fields.
  4. Every semisimple group is a direct product of modules.
Question 13 Multiple Choice (Single Answer)

What is the Burnside theorem?

  1. Every finite group of order $p^n$ is solvable.
  2. Every finite group of order $p^n$ is nilpotent.
  3. Every finite group of order $p^n$ is simple.
  4. Every finite group of order $p^n$ is perfect.
Question 14 Multiple Choice (Single Answer)

What is the Feit-Thompson theorem?

  1. Every finite group of odd order is solvable.
  2. Every finite group of odd order is nilpotent.
  3. Every finite group of odd order is simple.
  4. Every finite group of odd order is perfect.
Question 15 Multiple Choice (Single Answer)

What is the Tits alternative?

  1. Every finite group is either solvable or semisimple.
  2. Every finite group is either solvable or nilpotent.
  3. Every finite group is either solvable or simple.
  4. Every finite group is either solvable or perfect.
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