Let M2Z2 be the set of 2 times 2 matrices with entries in Z2

Let M_2(Z_2) be the set of 2 times 2 matrices with entries in Z_2. How many elements are in M_2(Z_2)? List all the elements. Note that GL_2(Z_2) = (A element GL_2(Z_2): det A 0 mod 2} and SL_2(Z_2) = (A element GL_2(Z_2): det A = 1 mod 2}. Why is SL_2(z_2 = GL_2(z_2)? List the element in this multiplication group. Let B = (1 1 1 0). Find (B) and |B|. Remember all operation are done mod 2. For example, (8 10 13 -1) = (0 0 1 1) Let G_1 = Z_1 times Z_5. We can extend the idea of generators of z_n to Z_n times Z_in, component-wise. Then ((a, b)) = {k(a, b): k element Z} = {(ka mod n, kb mod m): k element Z}. Find ((3, 3)). What is the order of (3, 3)? Let Ga_2 = U(4) times U(5). Likewise, we can extend the idea of generators of U(n) to U(n) times U(m) component-wise. Then ((a, b)) = {(a, b)^k: k element Z) = {(a^k mod n, b^k mod m): k element Z}. Find {(3.3)). What is the order of (3.3)?

Solution

Since there are 2 elements per entry in the 2x2 matrix,

|M2(Z2)| = 2⁴ = 16.

2)

Such a matrix must have nonzero determinant.

By enumeration, there are 6 such matrices:

[1 0].[1 0].[0 1].[0 1].[1 1].[1 1]

[1 1],[0 1],[1 1],[1 0],[0 1],[1 0].

3)

Take repeated powers until you get the identity (matrix).

[1 1][1 1]....[0 1]

[1 0][1 0].=.[1 1].

[1 1][0 1]....[1 0]

[1 0][1 1].=.[0 1]; so the given element has order 3.

4)

What is B? This should be a direct computation.