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. Every right coset Ha has the same number of elements as H.

    Circle one:   True   False

  2. Which equation must hold for a homomorphism between groups written multiplicatively?

    • f(ab) = f(a) + f(b)
    • f(ab) = f(a)f(b)
    • f(ab) = f(a) - f(b)
    • f(ab) = f(a)/f(b)

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

    Answer: ______________

  4. In additive Z4, H = {0, 2}. Which list is the full partition into distinct left cosets?

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

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

    Answer: ______________

  6. In additive Z8, H = {0, 4}. How many distinct left cosets does H have?

    Answer: ______________

  7. A group homomorphism with kernel consisting only of the source identity is injective.

    Circle one:   True   False

  8. In additive Z10, the coset 3 + H is {3, 8}. What is H?

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

  9. Let f(x) = x^2 on nonzero real numbers under multiplication. Is f a homomorphism?

    • Only negative inputs can satisfy f(ab) = f(a)f(b)
    • No, because squaring a nonzero real number gives zero
    • Yes, because f(ab) = f(a)f(b)

  10. A finite group has order 7. A student claims it has a subgroup of order 3. What rules this out?

    • 3 does not divide the group order 7
    • A subgroup must have more elements than its parent group
    • Every smaller positive integer is a possible subgroup order
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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 · Right multiplication by a is a bijection from H to Ha.
  2. f(ab) = f(a)f(b) · The target operation must combine the two mapped elements in the same way.
  3. 3 · Translation is a bijection, so each coset has H's three elements.
  4. {0, 2}, {1, 3} · H and 1 + H are disjoint and cover all four residues. Other representatives repeat these cosets.
  5. 3 · Among residues 0 through 11, the multiples of 4 are 0, 4 and 8.
  6. 4 · Each left coset has two elements, so eight residues form four cosets.
  7. True · Equal images force the quotient of the inputs into the trivial kernel, so the inputs are equal.
  8. {0, 5} · Subtracting the representative 3 recovers {0, 5}.
  9. Yes, because f(ab) = f(a)f(b) · Squaring stays within the nonzero reals and preserves multiplication.
  10. 3 does not divide the group order 7 · Lagrange's theorem requires a subgroup's order to divide 7.
Worksheet · LightMySky