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

One rule for many symmetries

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

  1. Which of these sets with the given operation forms a group?

    • Positive integers under addition
    • Integers under addition
    • Integers under subtraction

  2. How many elements sit in the symmetry group of a square?

    • 4
    • 6
    • 8

  3. Why do positive integers fail as a group under addition?

    • Addition is not associative there
    • There is no identity and no inverses inside the set
    • Sums escape the set

  4. Which of these sets with the given operation forms a group?

    • Integers under addition
    • Positive integers under addition
    • Integers under subtraction
    • Even integers under division

  5. How many symmetries does a square have, and what are they?

    • 8: four rotations and four reflections
    • 4: four rotations only
    • 6: three rotations and three reflections
    • 2: the identity and one flip

  6. Integers under subtraction form a group because differences of integers stay integers.

    Circle one:   True   False

  7. Nina checks that the even integers contain 0 and that the sum and the negative of evens are even, then concludes they form a subgroup of the integers under addition. Is Nina right?

    Circle one:   True   False

  8. Lee says the integers modulo 6 form a group under addition. Is Lee right?

    Circle one:   True   False

  9. A student claims integers modulo 6 form a group under both addition and multiplication. What is wrong?

    • Nothing is wrong
    • Addition works but multiplication lacks inverses
    • Multiplication works but addition fails

  10. In the symmetry group of the square, what is the order of a 90 degree rotation?

    • 4
    • 2
    • 8
    • 1
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Answer key

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

One rule for many symmetries W1-mt_HFAhcaGo2L-s1

  1. Integers under addition · Integers under addition meet all four axioms with identity zero and negatives as inverses.
  2. 8 · Four rotations plus four reflections make eight symmetries.
  3. There is no identity and no inverses inside the set · Zero is missing so there is no identity, and negatives are missing so there are no inverses.
  4. Integers under addition · The integers under addition satisfy all four axioms: closure, associativity, identity 0, and negatives as inverses.
  5. 8: four rotations and four reflections · The square has 8 symmetries: rotations through 0, 90, 180 and 270 degrees plus reflections across 4 axes.
  6. False · Closure alone is not enough: subtraction is not associative, so the axioms fail.
  7. True · Nina is right. She verified nonemptiness, closure under addition, and closure under inverses.
  8. True · Lee is right. Addition mod 6 is closed and associative, 0 is the identity, and each element has an inverse mod 6.
  9. Addition works but multiplication lacks inverses · Mod 6 addition cycles cleanly with identity 0, while most elements have no multiplicative inverse.
  10. 4 · Four quarter turns return the square to its start, and no smaller positive number does, so the order is 4.
Worksheet · LightMySky