Final answer:
In the group G of invertible 2x2 matrices with entries in Z₃, the operation is the usual matrix multiplication modulo 3. To determine if (G,⋅) is a group, we need to verify the four group axioms.
Step-by-step explanation:
In the group G of invertible 2x2 matrices with entries in Z₃, the operation is the usual matrix multiplication modulo 3. To determine if (G,⋅) is a group, we need to verify the four group axioms: closure, associativity, identity, and inverse.
Closure: To show closure, we need to demonstrate that the product of any two matrices in G is also in G. Since matrix multiplication is closed under the set of 2x2 matrices, this condition is satisfied.
Associativity: Matrix multiplication is associative, which means for any three matrices A, B, and C in G, (AB)C = A(BC). Therefore, the operation (⋅) is associative in G.
Identity: The identity matrix, I, serves as the identity element in matrix multiplication. If A is any matrix in G, then AI = IA = A. Therefore, the operation (⋅) has an identity element in G.
Inverse: To show the existence of inverses in G, we need to demonstrate that for every matrix A in G, there exists a matrix A' in G such that AA' = A'A = I. Invertible matrices have unique inverses, so this condition is satisfied.
Since G satisfies all four group axioms, (G,⋅) is indeed a group.