Count symmetries with orbits

Mathematics · Abstract Algebra · ages 22-23

Name ______________________ Date ____________

Rotations of a cube send any face to any of the 6 faces, and 4 rotations hold a chosen face fixed. What is the order of the rotation group?
- 24
- 10
- 12
- 20
Cube rotations move any face to any of 6 faces, and 4 spins fix one face. What is the rotation group order?
- 24
- 12
- 20
Mira claims a group with 27 elements must hold a central element besides the identity. Is she right?
Circle one: True False
A group of order 12 acts, and one point stabiliser has size 3. Divide to get the orbit size. What is it?
Answer: ______________
A tetrahedron has 4 faces with 3 spins fixing each one. How many rotational symmetries does it have?
Answer: ______________
A cube has 8 vertices, and 3 rotations fix a given vertex (spins about its body diagonal). What is the rotation group order?
- 24
- 11
- 16
- 32
Mira claims a group with 27 elements must have a central element besides the identity. Is she right?
Circle one: True False
A group of order 20 acts transitively on 5 objects. How many of its elements fix one chosen object?
Answer: ______________
Why must a group of prime power order have a nontrivial center?
- Every element commutes in such groups
- Each larger class size shares the prime factor, forcing the center past the identity
- The class equation never applies to such groups
A friend claims the stabiliser of a point is the set of points the group moves. What is wrong?
- Nothing is wrong, that description is exact
- Counting never needs the group law
- The stabiliser is the subgroup of elements fixing the point, not points moved

Answer key

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

Count symmetries with orbits W1-mt_urut0TFTOR-s1

24 · Orbit size 6 times stabiliser size 4 gives 24 rotations.
24 · Orbit size 6 times stabiliser size 4 gives 24 rotations.
True · The class equation forces the center past the identity alone.
4 · Orbit size is group order over stabiliser size, so 12 over 3 is 4.
12 · Four faces times a stabiliser of 3 gives 12 symmetries.
24 · Eight vertices times a stabiliser of 3 gives 24, matching the face count.
True · The class equation forces the centre of a p-group past the identity alone.
4 · One orbit of size 5 leaves stabilisers of 20 over 5, or 4.
Each larger class size shares the prime factor, forcing the center past the identity · The shared prime factor leaves the center holding more than the identity.
The stabiliser is the subgroup of elements fixing the point, not points moved · Stabilisers collect fixing elements, while orbits collect visited points.