Answer:
To show that the set {1, -1} is an Abelian group under multiplication, we need to verify that it satisfies the four properties of a group and the additional property of commutativity (Abelian property).
A group is defined by the following properties:
1. Closure: For all elements a, b in the group, the product ab is also in the group.
2. Associativity: For all elements a, b, c in the group, (ab)c = a(bc).
3. Identity element: There exists an element e in the group such that for any element a in the group, ae = ea = a.
4. Inverse element: For every element a in the group, there exists an element b in the group such that ab = ba = e (the identity element).
Now, let's check these properties for the set {1, -1} under multiplication:
1. Closure: The multiplication of 1 and -1 is -1, which is in the set. The set is closed under multiplication.
2. Associativity: Since there are only two elements in the set, associativity is trivially satisfied.
3. Identity element: The element 1 serves as the identity element because 1 * a = a * 1 = a for any element a in the set.
4. Inverse element: Both 1 and -1 are their own inverses, as 1 * 1 = 1 and -1 * -1 = 1.
5. Commutativity (Abelian property): For all elements a, b in the set, ab = ba because multiplication is commutative.
Since all the group properties are satisfied, and the set also satisfies the commutativity property (Abelian property), we can conclude that the set {1, -1} forms an Abelian group under multiplication.