Group Theory : Permutation Groups and Even Permutations

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 )

Group Theory : Permutation Groups

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

Group Theory: Permutation Groups

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

Group Theory : Permutation Groups

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

Group Theory : Cycles and Permutation Groups

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

Group Theory : Permutation Groups

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

Group Theory : Permutation Groups

Given the permutation x = ( 1 2 3 ), y = ( 1 3 5) find a permutation a such that a^( – 1) x a = y

Group Theory : Conjugates and Permutation Groups

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

Group Theory: 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.

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.

Group Theory : Find the number of conjugates of (1 2)(3 4) in Sn , n ≥ 4.

Find the number of conjugates of (1 2)(3 4) in Sn , n ≥ 4.