1. Label the figure as shown. Clearly (1 3) and (2 4) are motions, as is their product. Hence the group of motions is {
ε, (13), (24), (13)(24)}, isomorphic to the Klein group
K4.
3. Label the figure as shown. Then (123) and (132) are motions (rotations of 120
and 240
about a line through vertex 4 and the center of the triangle base). Clearly every motion (indeed every symmetry) must fix vertex 4. Hence the group of motions is
G = {
ε, (123), (132)}. However (12), (13) and (23) are all symmetries (which are not motions), so the group of symmetries if
S3.
5. Label the figure as shown. Clearly (12)(34) and (14)(23) are such symmetries, and hence their product is (13)(24). The rest of the symmetries of the square do not preserve blue edges, so the group is {
ε, (12)(34), (13)(24), (14)(23)} ≅
K4.
7. Label the vertices as shown. Let
λ = (1 3)(2 4)(5 7)(6 8) and
μ = (1 6)(2 5)(3 8)(4 7). These are motions (rotations of
π radians about axes through the sides). Also
λμ =
μλ = (1 8)(2 7)(3 6)(4 5) is the sides). Also
λμ =
μλ = (1 8)(2 7)(3 6)(4 5) is the rotation about a vertical axis. The group of motions is {
ε,
λ,
μ,
λμ} ≅
K4. However, there are symmetries which are not motions. We have
which are reflections in various planes of symmetry. Now compute
Call these last two
γ =
λτ and
δ =
λσ. Then the group of symmetries is
The fact that
x2 =
ε for all
x G gives the following multiplication table:
The group is abelian and x2 = ε for all x. These are called elementary abelian groups.