Group Actions, Orbits and the Class Equation · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Count symmetries with orbits

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

  1. S3 acts on {1, 2, 3} by permuting the letters. Which elements form the stabiliser of 1?

    • The identity and the swap of 1 with 2
    • The identity and the swap of 1 with 3
    • The identity and the swap of 2 with 3
    • The two swaps involving 1, without the identity

  2. Cube rotations move a face to any of 6 faces. Exactly 4 rotations preserve a chosen face as a set, including the identity. What is the rotation group order?

    • 24
    • 12
    • 20

  3. When a group acts on itself by conjugation, the orbit of x is its conjugacy class.

    Circle one:   True   False

  4. A group of order 12 acts, and one point stabiliser has size 3. Divide to get the orbit size. What is it?

    Answer: ______________

  5. A regular tetrahedron has 4 faces. Exactly 3 rotations preserve a chosen face as a set, including the identity. How many rotational symmetries does it have?

    Answer: ______________

  6. G = {e, s} acts on {1, 2, 3}. The element s swaps 1 and 2 and fixes 3. Which list gives all its orbits?

    • The sets {1, 2} and {3}
    • The sets {1, 3} and {2}
    • The sets {2, 3} and {1}
    • The single set {1, 2, 3}

  7. If a nonidentity element x has a singleton conjugacy class, then x is central, even when the group order is not a prime power.

    Circle one:   True   False

  8. A group of order 20 acts transitively on 5 objects. How many of its elements fix one chosen object?

    Answer: ______________

  9. Why must a group of order p^n, with p prime and n at least 1, have a nontrivial centre?

    • Every such group is abelian, so all its conjugacy classes contain exactly one element
    • Noncentral class sizes are divisible by p, so the positive centre size is too
    • Noncentral class sizes never divide the group order, so the class equation does not apply

  10. A friend calls all points reachable from x its 'stabiliser'. Which correction fits?

    • The stabiliser contains reachable points; the orbit contains elements fixing x
    • The stabiliser contains all group elements; the orbit contains only the point x
    • The stabiliser contains elements fixing x; the orbit contains reachable points
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Answer key

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

Count symmetries with orbits W1-mt_urut0TFTOR-s1

  1. The identity and the swap of 2 with 3 · Fixing 1 leaves two possibilities: fix both other letters or exchange 2 and 3.
  2. 24 · Orbit size 6 times stabiliser size 4 gives 24 rotations.
  3. True · Both sets consist of the elements gxg^-1 as g ranges over the group.
  4. 4 · Orbit size is group order over stabiliser size, so 12 over 3 is 4.
  5. 12 · Four faces times a stabiliser of 3 gives 12 symmetries.
  6. The sets {1, 2} and {3} · The first two points can reach each other. The point 3 is fixed by both group elements, so it forms a separate orbit.
  7. True · A singleton class gives gxg^-1 = x for every g. Multiplying by g on the right yields gx = xg.
  8. 4 · With one orbit of size 5, the stabilisers have size 20 divided by 5, or 4.
  9. Noncentral class sizes are divisible by p, so the positive centre size is too · Each noncentral class size is divisible by p. The class equation makes the centre size divisible by p; since it contains the identity, its size is at least p.
  10. The stabiliser contains elements fixing x; the orbit contains reachable points · Stabilisers collect fixing elements, while orbits collect visited points.
Worksheet · LightMySky