Final answer:
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 *.