Group Representations and the Group Algebra · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Groups acting as matrices

Mathematics · Abstract Algebra · ages 22-23
Name ______________________   Date ____________

  1. Symmetric group on 3 letters. What is the dimension of its regular representation?

    Answer: ______________

  2. Cyclic group of order 2. Which matrices give a faithful representation?

    • {[1], [1]}
    • {[0], [0]}
    • {[2], [3]}
    • {[1], [-1]} as 1 by 1

  3. In a representation, what does each group element become?

    • A single number with no structure
    • An invertible linear map on a vector space
    • A subspace of the group

  4. A nonzero representation with no proper nonzero subrepresentations is called irreducible.

    Circle one:   True   False

  5. A representation is the same as a module over the group algebra, with the group acting linearly. How does a representation give a linear action?

    • Each element acts as an invertible linear map on a vector space
    • Each element acts as a plain number
    • The action is always trivial

  6. A nonzero representation with no proper nonzero subrepresentations is called irreducible. Is this correct?

    Circle one:   True   False

  7. How does a representation give a linear action?

    • Each group element acts as a number
    • Each group element acts as an invertible linear map on a vector space
    • Action is always trivial
    • No action exists

  8. The symmetric group on 3 letters has 6 elements. Its regular representation has dimension equal to the group order. Type that dimension.

    Answer: ______________

  9. A 3 dimensional representation has a 1 dimensional subspace preserved by the whole group. What follows?

    • It is irreducible, because 1 is small
    • It is faithful, because dimensions divide
    • It is reducible, with a 1 dimensional subrepresentation

  10. Rotation by a quarter turn has order how much?

    Answer: ______________

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Answer key

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Groups acting as matrices W1-mt_y5JlvQGKYW-s1

  1. 6 · Regular dimension equals group order, and symmetric group on 3 letters has 6 elements, so dimension is 6.
  2. {[1], [-1]} as 1 by 1 · {[1], [-1]} as 1 by 1 is faithful since squares match the group law, while the constant, zero, and [2], [3] versions break homomorphism or faithfulness.
  3. An invertible linear map on a vector space · Each element acts as an invertible linear map, realized as an invertible matrix.
  4. True · That is exactly the definition: no smaller stable piece inside.
  5. Each element acts as an invertible linear map on a vector space · The matrices of the representation are exactly the linear maps of the action.
  6. True · This is the definition of irreducible.
  7. Each group element acts as an invertible linear map on a vector space · Each group element acts as an invertible linear map on a vector space is the definition, while the others deny structure.
  8. 6 · The regular representation has dimension equal to the group order, which is 6.
  9. It is reducible, with a 1 dimensional subrepresentation · A proper nonzero stable subspace is a subrepresentation, which rules out irreducibility.
  10. 4 · Four quarter turns make a full turn, so order is 4.
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