Group Theory (XC): Determine for what m an m-cycle is an even permutation.

Modern Algebra Group Theory (XC) Permutation Groups Even Permutation Determine for what m an m-cycle is an even permutation.

Group Theory (XCI): Determine which of the following are even permutations: (a) ( 1 2 3 ) (b) ( 1 2 3 4 5 )( 1 2 3 )( 4 5 ) (c ) ( 1 2 )( 1 3 )( 1 4 )( 2 5 )

Modern Algebra Group Theory (XCI) Permutation Groups Even Permutation Determine which of ...continues

Group Theory (XCII): Prove that the smallest subgroup of Sn containing ( 1 , 2 )and ( 1 , 2 , … , n ) is Sn. ( In other words, these generate Sn )

Modern Algebra Group Theory (XCII) Permutation Groups The Symmetric Group of Degree n Prove that the smallest subgroup of Sn containing ...continues

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

Modern Algebra Group Theory (XCIII) Permutation Groups To find a permutation a such that a^( – 1) x a = y ...continues

Group Theory (XCIV): Given the permutation x = ( 1 2 )( 3 4 )( 5 6 ), y = ( 1 3 )( 2 5 )( 4 6 ) find a permutation a such that a^( – 1) x a = y.

Modern Algebra Group Theory (XCIV) Permutation Groups To find a permutation a such that a^( – 1) x a = y Given the permutation x ...continues

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). ...continues

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 ...continues

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 ...continues

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) ...continues

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 , … , ...continues