Normal Subgroups and Quotient Groups · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

When cosets form a group

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

  1. Cyclic group of order 6 has a subgroup of order 2. What is the order of the quotient?

    Answer: ______________

  2. Symmetric group on 3 letters has order 6. Alternating subgroup has order 3. What is its index and is it normal?

    • Index 3, normal
    • Index 2, not normal
    • Index 2, normal
    • Index 3, not normal

  3. The permutations of 3 letters have order 6, and the alternating subgroup has order 3. What are its index and normality?

    • Index 3 and normal
    • Index 2 but not normal
    • Index 2 and normal

  4. Every subgroup of an abelian group is normal, because conjugation changes nothing there.

    Circle one:   True   False

  5. A map from the cyclic group of order 6 onto the cyclic group of order 3 has kernel of order 2, and its image matches the quotient by that kernel.

    Circle one:   True   False

  6. Homomorphism from cyclic of order 6 onto cyclic of order 3 has kernel of order 2, image is whole codomain. Is the image isomorphic to the quotient by the kernel?

    Circle one:   True   False

  7. Subgroup generated by the transposition swapping 1 and 2 inside permutations of 3 letters. Is it normal?

    • No, conjugating by the swap of 1 and 3 gives the swap of 2 and 3 outside
    • Yes, all subgroups are normal
    • Yes, because index is 3
    • No, because the group is abelian

  8. Is the subgroup generated by the swap of 1 and 2 normal inside the permutations of 3 letters?

    • No, conjugating by the swap of 1 and 3 escapes it
    • Yes, every subgroup there is normal
    • Yes, because its index is 3

  9. The sign map sends permutations of 3 letters to 1 and minus 1. What is its image?

    • Only the value 1
    • Both values 1 and minus 1
    • Only the alternating subgroup

  10. Dihedral group of the square has 8 elements. Its center has 2 elements. What is the order of the quotient?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

When cosets form a group W1-mt_RU7A-t7wQ5-s1

  1. 3 · Quotient order equals 6 divided by 2, which is 3.
  2. Index 2, normal · Index 2, normal is correct since 6 divided by 3 equals 2, and index 2 subgroups are always normal.
  3. Index 2 and normal · 6 divided by 3 is 2, and index 2 always forces normality.
  4. True · Commutativity collapses each conjugate back to the element itself.
  5. True · This is the isomorphism theorem working on a concrete map.
  6. True · This is the isomorphism theorem: image is isomorphic to domain modulo kernel.
  7. No, conjugating by the swap of 1 and 3 gives the swap of 2 and 3 outside · No, conjugating by the swap of 1 and 3 gives the swap of 2 and 3 outside is correct, showing nonnormality, while the group is nonabelian and index 3 does not force normality.
  8. No, conjugating by the swap of 1 and 3 escapes it · The conjugate swap of 2 and 3 lies outside, so normality fails.
  9. Both values 1 and minus 1 · Both values occur, so the image is the full pair.
  10. 4 · 8 divided by 2 equals 4.
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