Permutations and Disjoint Cycles

Let b be the permutation (1 2 3)(4 5 6 7)(8 9 10 11 12 13) what is b^99 as a product of disjoint cycles. -I know b^99=b^3 but I'm a little confused on the disjoint cycles part.

Combinations, Permutations and Cycles

How many elements of order 5 are there in S_8? - I think there are 8! / 3!5! = 56 ways to order the elements in the cycle but how many of order 5 are there? keywords: S8

Groups and Matrices

1 3 1 4 Let the matrix A = 0 2 and the matrix B = 5 1 be elements in GL(2, Z_7). Find (A^-1 * B^-1)^-1. - I am unsure of when to perform the operation mod 7.

Group Elements : List all of the elements of order 15 in Z_600

List all of the elements of order 15 in Z_600.

Cyclic Groups and Generators

Let G = be a cyclic group of size 600. What is a generator for the smallest subgroup of G that includes both a^42 and a^70?

Symmetric Groups and Disjoint Cycles

Suppose that τ in Sn fixes no symbol. Show that τ = μ^m for some n-cycle μ and positive integer m if and only if τ is the product of disjoint cycles of equal length. I know that τ can be written as the product of disjoint cycles, but am not sure how to proceed from there. See attached file fo ...continues

Abelian Groups and Relatively Prime Order

Suppose G is an abelian group of order m. Let m, n be relatively prime. Show that for every element g in G, there exists an element x in G such that x^m=g.

Finite abelian groups all of whose elements (except the identity element) are of the same order

Give examples of finite abelian groups in which all elements (except the identity element) are of the same order.

Mappings, Homomorphisms and Subgroups

Let @:G-->H be a homomorphism of G onto H, and let N be a normal subgroup of G. Show that @(N) is a normal subgroup of H. How do I prove that a mapping is a normal subgroup of a group? What I am missing here is some understanding of the terminology and some clear understanding of mappings, homomorphisms, subgroups and no ...continues

Subgroups and indexes

Explain what the index of a subgroup and a coset of a group are. Also, prove that if N is a subgroup of a group G such that [G: N] = 2, and if "a" and "b" are elements of G, then the product "ab" is an element of N if and only if either (1) both "a" and "b" are elements of N or (2) neither "a" nor "b" is an element of N.