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. In additive Z12, H = {0, 4, 8}. Label cosets by representatives 0, 1, 2, 3. What representative labels the entry in row 3 and column 1 of the quotient addition table?

    Answer: ______________

  2. N is the kernel of a homomorphism phi: G -> K. For n in N, which calculation proves gng^-1 is also in N?

    • phi(gng^-1) = phi(g), which need not be the identity
    • phi(gng^-1) = phi(n)phi(g), which need not be the identity
    • phi(gng^-1) = phi(g)e phi(g)^-1 = e
    • phi(gng^-1) = gng^-1, because every homomorphism fixes inputs

  3. The permutations of 3 letters have order 6, and the alternating subgroup has order 3. What is its index, and is it normal?

    • 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 group homomorphism from the cyclic group of order 6 onto the cyclic group of order 3 has kernel of order 2, and its image is isomorphic to the quotient by that kernel.

    Circle one:   True   False

  6. The homomorphism phi: Z4 -> Z8 sends a to 2a and has kernel {0}. The first isomorphism theorem therefore makes Z4 isomorphic to the whole codomain Z8.

    Circle one:   True   False

  7. In additive Z12, H = {0, 6}. Which coset equals 3 + H?

    • 9 + H = {3, 9}
    • 1 + H = {1, 7}
    • 2 + H = {2, 8}
    • 6 + H = {0, 6}

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

    • No: conjugating by (13) sends (12) outside the subgroup
    • Yes: conjugating by (13) keeps (12) inside the subgroup
    • Yes: every subgroup of index 3 is normal in its group

  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. In additive Z18, H = {0, 6, 12}. What is the order of the element 4 + H in Z18/H?

    Answer: ______________

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Answer key

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

  1. 0 · The entry is (3 + H) + (1 + H) = 4 + H = H, labelled 0.
  2. phi(gng^-1) = phi(g)e phi(g)^-1 = e · The homomorphism preserves products and inverses. Since phi(n) = e, the conjugate maps to e and belongs to the kernel.
  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. False · The image is {0, 2, 4, 6}. The theorem identifies Z4 with this image, not with the eight-element codomain.
  7. 9 + H = {3, 9} · 3 + H is {3, 9}. Translating H by 9 gives the same set.
  8. No: conjugating by (13) sends (12) outside the subgroup · 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. 3 · The first positive multiples of 4 are 4, 8 and 12. The third is the first in H, so the coset has order 3.
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