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.
1) This is a simple counting problem, unworthy of anyone studying group theory. How many possibilities are there for a? What about b? how many possible combinations are there?
2) The only way to do this (as far as I know) is to go through each of the properties of a group in the definition, one by one. This may take a while but none of them are too hard, although if you haven't used this method before, you may need some help. Have a go and post if you have any problems
3)A multiplication table lists all the elements of the group across the top and left side and the products of the row and column where they meet, much like this, which you probably saw in primary school. You can do this. Believe in yourself, give it a go and you will be surprised at what you can do.
4)Don't look at this one until you have done the others.