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2. Let's do some linear algebra over a finite field Z2, the integers modulo 2. (Use just the numbers 0 and 1.) The set of 2 x 2 matrices y = (a b / c d) having as entries integers modulo 2 (that is, a, b, c, d in Z2) and with determinant det y = 1 is denoted SL(2, Z2). (a) Show that SL(2, Z2) is a finite group. Write down all of its elements and their orders. (Hint: there are just 6 of them.) Identify this group from its multiplicative structure. (b) How many distinct non-zero row vectors x = (s, t) are there in the vector space Z2^2? Denote this set by P(Z2) = Z2^2 - {0}. (c) Consider SL(2, Z2), using matrix multiplication on the right; that is, x * y = (s t)(a b / c d) = (s1 t1) say, acting as linear transformations on the vector space Z2^2. Show that SL(2, Z2) is the complete group of permutations of the row vectors in P(Z2).

          2. Let's do some linear algebra over a finite field Z2, the integers modulo 2. (Use just the numbers 0 and 1.) The set of 2 x 2 matrices y = (a b / c d) having as entries integers modulo 2 (that is, a, b, c, d in Z2) and with determinant det y = 1 is denoted SL(2, Z2). (a) Show that SL(2, Z2) is a finite group. Write down all of its elements and their orders. (Hint: there are just 6 of them.) Identify this group from its multiplicative structure. (b) How many distinct non-zero row vectors x = (s, t) are there in the vector space Z2^2? Denote this set by P(Z2) = Z2^2 - {0}. (c) Consider SL(2, Z2), using matrix multiplication on the right; that is, x * y = (s t)(a b / c d) = (s1 t1) say, acting as linear transformations on the vector space Z2^2. Show that SL(2, Z2) is the complete group of permutations of the row vectors in P(Z2).

2. Let's do some linear algebra over a finite field Z2, the integers modulo 2. (Use just the numbers 0 and 1.) The set of 2 x 2 matrices y = (a b / c d) having as entries integers modulo 2 (that is, a, b, c, d in Z2) and with determinant det y = 1 is denoted SL(2, Z2). (a) Show that SL(2, Z2) is a finite group. Write down all of its elements and their orders. (Hint: there are just 6 of them.) Identify this group from its multiplicative structure. (b) How many distinct non-zero row vectors x = (s, t) are there in the vector space Z2^2? Denote this set by P(Z2) = Z2^2 - 0. (c) Consider SL(2, Z2), using matrix multiplication on the right; that is, x * y = (s t)(a b / c d) = (s1 t1) say, acting as linear transformations on the vector space Z2^2. Show that SL(2, Z2) is the complete group of permutations of the row vectors in P(Z2).

Added by David S.

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