Group Theory (LXXXII): Prove that ( 1, 2, 3, …, n )^(-1) = ( n, n – 1, n – 2, …, 3, 2, 1 )

Modern Algebra Group Theory (LXXXII) Permutation Groups The Inverse of a Cycle The Inverse of a Permutation Prove that ( 1, 2, 3, …, n )^(-1) = ( n, n – 1, n – 2, …, 3, 2, 1 )

Group Theory (LXXXIII): Find the cycle structure of all the powers of ( 1, 2, 3, …, 8 )

Modern Algebra Group Theory (LXXXIII) Permutation Groups The Cycle structure of all the Powers of a Permutation Find the cycle structure of all the powers of ( 1, 2, 3, …, 8 )

Group Theory (LXXXIV): (a) What is the order of an n-cycle? (b) What is the order of the product of the disjoint cycles of lengths m1 , m2 , … , mk ? (c) How do you find the order of a given permutation?

Modern Algebra Group Theory (LXXXIV) Permutation Groups The Order of a Permutation The Order of a Cycle (a) What is the order of an n-cycle? (b) What is the order of the product of the disjoint cycles of lengths m1 , m2 , … , mk ? (c) How do you find the order of a given permutation?

Group Theory (LXXXV): Compute a^( – 1 )ba where a = ( 1 3 5 )( 1 2 ) b = ( 1 5 7 9 )

Modern Algebra Group Theory (LXXXV) Permutation Groups The Computation of a^( – 1 )ba Compute a^( – 1 )ba where a = ( 1 3 5 )( 1 2 ) b = ( 1 5 7 9 )

Group Theory (LXXXVI): Compute a^( – 1 )ba where a = ( 5 7 9 ) b = ( 1 2 3 )

Modern Algebra Group Theory (LXXXVI) Permutation Groups The Computation of a^( – 1 )ba Compute a^( – 1 )ba where a = ( 5 7 9 ) b = ( 1 2 3 )