Consider the set G of all 2×2 non-zero matrices of the
form (a b )
.......(b a + b)
where a, b belong to Z3, the set of congruence classes modulo 3.
(1) Find the number of elements in the set G.
(2) Prove that G is a group under matrix multiplication modulo 3.
(3) Complete the multiplication table for G.
(4) Is the group G isomorphic to the group of symmetries of a square?
(two groups are isomorphic if there exists a bijection
between them that preserves the group operations, in other words,
after an appropriate permutation and relabeling of the elements,
their multiplication tables are identical.