Homomorphisms, Cosets and Lagrange's Theorem · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Maps, slices, and counting

Mathematics · Abstract Algebra · ages 21-22
Name ______________________   Date ____________

  1. True or false: any two cosets of the same subgroup are either exactly the same set or share no elements at all.

    Circle one:   True   False

  2. φ sends each integer n to n mod 5, giving a homomorphism from (Z, +) to (Z₅, +). Which of these numbers belongs to the kernel of φ?

    • 5
    • 3
    • 7
    • 2

  3. In Z₉, let H = {0, 3, 6}. How many elements are in each coset of H?

    Answer: ______________

  4. In Z₆, let H = {0, 3}. What is the coset 2 + H?

    • {2, 5}
    • {2, 3}
    • {0, 3}
    • {1, 4}

  5. The map sends Z12 to remainders mod 4. How many elements land on 0?

    Answer: ______________

  6. In Z8, H equals 0 and 4. How many distinct cosets does H have?

    Answer: ______________

  7. Let φ(x) = x² on the group of nonzero real numbers under multiplication. True or false: φ is a homomorphism.

    Circle one:   True   False

  8. In Z₁₀, let H = {0, 5}. What is the coset 3 + H?

    • {3, 8}
    • {3, 5}
    • {0, 5}
    • {5, 8}

  9. Let f of x equal x squared on nonzero reals under multiplication. Is f a homomorphism?

    • Only for negative numbers
    • No, since squares leave the set
    • Yes, since (a b) squared equals a squared times b squared

  10. A group has order 7. A student claims it holds a subgroup of order 3. What is wrong?

    • 3 does not divide 7, so Lagrange rules it out
    • Subgroups must be larger than the group
    • Order 3 always fits
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Answer key

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

Maps, slices, and counting W1-mt_yKiZ8sg7yN-s1

  1. True · Cosets slice a group into equal-sized, non-overlapping pieces, which is exactly what makes counting them work for Lagrange's theorem.
  2. 5 · A number is in the kernel when φ sends it to 0, the identity of Z₅.
  3. 3 · Every coset of a subgroup has the same number of elements as the subgroup itself.
  4. {2, 5} · A coset is formed by adding the same element to every member of H.
  5. 3 · Remainders 0 come from 0, 4, and 8, which is 3 elements.
  6. 4 · Cosets match H in size, so 8 divided by 2 gives 4.
  7. True · A map is a homomorphism when it preserves the operation: φ(a·b) must equal φ(a)·φ(b).
  8. {3, 8} · Add the representative to every element of H to build the coset.
  9. Yes, since (a b) squared equals a squared times b squared · Both routes give a squared times b squared, so the operation is preserved.
  10. 3 does not divide 7, so Lagrange rules it out · Subgroup orders must divide the group order, and 3 does not divide 7.
Worksheet · LightMySky