In exercises 1-4, tell whether or not the statement is true for all groups of transformations with at least two elements.

1. The number of elements is infinite.

Ans:False. Solution: For example the group of transformations of an equilateral triangle is a finite group.

2.The product of each two elements is a transformation.

Ans:True.

3. Each group includes an identity element.

Ans: True.

4. The commutative property holds for all the elements of the group.

Ans: False. Solution: This is not true for the group of transformations of an equilateral triangle, for instance.

5. Using only the other three properties, prove that a group of transformations includes the identity element.

Solution: Let g be an element of the group. By the 4th property, there exists an element g^-1 in the group. Now, by property 1, g(g^-1) is also in the group. But, g(g^-1) = the identity element of the group by property 4.

For exercises 7-10, find the products from the multiplication table for the symmetries of an equilateral triangle, then use the permutation notation to verify each answer. We will use cyclic notation to answer these questions. R₁ = (2 3), R₂ = (1 3), R₃ = (1 2), R(120) = (1 2 3), and R(240) = (1 3 2).

7. R(240)R₃

Solution:R₁. The product R(240)R₃, in cyclic notation is, (1 3 2)(1 2) = (1)(2 3) = (2 3).

8. R₃R(240)

Solution:R₂. The product R₃R(240), in cyclic notation is, (1 2) (1 3 2) = (1 3)(2) = (1 3).

9. R₁R₃

Solution:R(240). The product R₁R₃, in cyclic notation is, (2 3)(1 2) = (1 3 2).

10. R₃R₁

Solution:R(120). The product R₃R₁, in cyclic notation is, (1 2) (2 3) = (1 2 3).

11. Is the set {I, R₁, R₂, R₃} of symmetries for the equilateral triangle a subgroup?

Ans:No. Solution: This set is not closed under composition, for instance the product R₁R₃ is not in the set. So, the set is not a group and therefore can not be a subgroup.

16. Prepare an operation table for the symmetries of a nonsquare rectangle.

Solution:

17. Verify that the symmetries of a nonsquare rectangle form a group.

Solution:The set satisfies closure because there are no empty cells in the above table. The operation is associative since it is composition of mappings. I is the identity element. The inverse of any element is itself, i.e., V^-1 = V since VV = I, for example.

18. List all subgroups of the group of symmetries of a nonsquare rectangle.

Solution: {I}, {I,V}, {I,H}, {I,R}, {I, V, H, R}

19. Which subgroups of the symmetries of a nonsquare rectangle are commutative groups?

Ans: All the subgroups are commutative.