MATH 113 Lecture Notes - Lecture 12: Alternating Group, Rhode Island Route 2, If And Only If
Document Summary
Observe if we have a cycle, (a1, a2, , an) = (a1, an) (a1, an-1) (a1, an-2) (a1, a2). Def: a cycle of length 2 is called a transposition. Then no permutation of sn can be expressed both as a product of an even number of transpositions and an odd number. Take (1, 2, 3, 4, 5) -> (a1, a2, a3, a4, a5). Express this using the rows of a matrix thus there are 0s and 1s, generally along the diagonal. Thus its determinant is either 1 or -1. If you switch any two rows, the rows resemble the positions replaced by the corresponding transposition. If you switch two columns, your determinant goes negative (or positive if you"re already negative). This means det(t1) is -1: a single transposition. A(original matrix) = t1* t2 * t3 * t4 * * tk. Det(a) is always 1 or -1, does not depend on the t1 and t2 and etc.
