Question

Let G be a multiplicative group. Let Gᵒᵖ be the set G equipped with a new operation * defined by a ∗ b = ba.

(a) Prove that Gᵒᵖ is a group.

Answer 1

To prove that Gᵒᵖ is a group, we verified that it satisfies all group axioms: closure, associativity, identity, and inverses, by using the properties of the original group G.

Step-by-step explanation:

The question is asking to prove that a set Gop with a new operation defined by a * b = ba is a group when G itself is given to be a multiplicative group. To establish that Gop is a group, we need to show that it satisfies the four group axioms: closure, associativity, identity, and inverses.

• Closure: For any elements a, b in Gop, a * b = ba is also an element of G since G is a group.
• Associativity: For any elements a, b, c in Gop, (a * b) * c = cba = a * (b * c). Thus, the operation * is associative.
• Identity: The identity element of Gop is the same as that of G, e, since for any a in Gop, a * e = ea = a and e * a = ae = a.
• Inverses: For every element a in Gop, its inverse a-1 in G is also its inverse in Gop, since a * a-1 = a-1a = e and a-1 * a = aa-1 = e.

Since all four axioms are satisfied, we can conclude that Gop is indeed a group under the operation *.