Group Theory (XCV): In Sn prove that there are (1/r). [n!/( n – r ) ] distinct r cycles.

Modern Algebra Group Theory (XCV) Permutation Groups Another Counting Principle In Sn prove that there are (1/r). [n!/( n – r ) ] distinct r cycles.

Group Theory (XCVI): Given the permutation x = ( 1 2 ), y = ( 3 4 ) find a permutation a such that a^( – 1) x a = y.

Modern Algebra Group Theory (XCVI) Permutation Groups To find a permutation a such that a^( – 1) x a = y Given the permutation x = ( 1 2 ), y = ( 3 4 ) find a permutation a such that a^( – 1) x a = y.

Group Theory (XCVII): Given the permutation x = ( 1 2 3 ), y = ( 1 3 5 ) find a permutation a such that a^( – 1) x a = y.

Modern Algebra Group Theory (XCVII) Permutation Groups To find a permutation a such that a^( – 1) x a = y Given the permutation x = ( 1 2 3 ), y = ( 1 3 5 ) find a permutation a such that a^( – 1) x a = y

Group Theory (XCVIII): Find the number of conjugates that the r-cycle (1 , 2 , … , r) has in Sn .

Modern Algebra Group Theory (XCVIII) Permutation Groups Another Counting Principle Find the number of conjugates that the r-cycle (1 , 2 , … , r) has in Sn .

Group Theory (XCIX): Prove that any element σ in Sn which commutes with (1 , 2 , … , r) is of the form σ = (1 , 2 , … , r)^i τ where i = 0, 1 , 2 , … , r , τ is a permutation leaving all of 1 , 2 , … , r fixed.

Modern Algebra Group Theory (XCIX) Permutation Groups Another Counting Principle Prove that any element σ in Sn which commutes with (1 , 2 , … , r) is of the form σ = (1 , 2 , … , r)^i τ where i = 0, 1 , 2 , … , r , τ is a permutation leaving all of 1 , 2 , … , r fixed.