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

Find the cycle structure of all the powers of ( 1, 2, 3, …, 8 )

Group Theory : Cycles and Permutation Groups

(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 : Permutation Groups

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

Group Theory : Permutation Groups

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

Group Theory : Permutation Groups

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

Groups: Conjugacy Classes

2.99 (iii) Show that there are two conjugacy classes of 5-cycles in A5, each of which has 12 elements.

Groups: Conjugacy Classes

(iv) Prove that the conjugacy classes in A5 have sizes 1, 12, 12, 15, and 20.

Group Theory : Permutation Groups

Prove that there is no a such that a^( – 1 ) ( 1 , 2 , 3 ) a = ( 1 , 3 )( 5 , 7 , 8 )

Group Theory : Permutation Groups

Prove that there is no permutation a such that a^( – 1 ) ( 1 , 2 ) a = ( 3 , 4 )( 1 , 5 )

Group Theory : Permutation Groups and Even Permutations

Determine for what m an m-cycle is an even permutation.