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Example: Suppose A = {1, 2, 3, 4} . We will denote the permutation that maps as follows: σ(1) = 3, σ(2) = 1, σ(3) = 4, σ(4) = 2 Study at Linköping University and get the competitive edge! Interested in Computer Science and connected fields? Kick-start your career with a master’s degree from Linköping University, Sweden – one of top 50 universities under 50 years old. com 55 Click on the ad to read more An Introduction to Abstract Algebra Group Theory 1 2 3 4 . The key to reading this notation is that the image of each of 3 1 4 2 the elements in the top row is located directly below.

6 permutations. We list the permutations of S3 below: ι= ρ1 = 1 2 3 1 2 3 1 2 3 1 3 2 , σ1 = , ρ2 = 1 2 3 2 3 1 1 2 3 3 2 1 , σ2 = 1 2 3 3 1 2 , ρ3 = 1 2 3 2 1 3 There are 62 = 36 possible pairings of these six permutations. Fortunately we do not have to calculate all of these. The ι ◦ µ = µ ◦ ι = µ for any permutaton µ . Furthermore, since each of the permutations labeled with ρ switch two elements of the set A , ρ21 = ρ22 = ρ23 = iι. The other compositions will take some work to figure out.

Subtraction is neither associative nor commutative on Z+. We see (2 − 3) − 4 = −5 = 3 = 2 − (3 − 4) , so it is not associative. The commutative property fails as well due to 2 − 3 = 3 − 2 . • Let Mn (R) denote n × n matrices with real entries. Matrix addition is commutative and associative, matrix multiplication is associative but not commutative. • Given real numbers x, y ∈ R,,Z, define N. a binary operation x · y = (x + y)2 . Since (x + y)2 = x2 + 2xy + y 2 = y 2 + 2yx + x2 = (y + x)2 = y · x this binary operation is commutative.