Direct product and isomorphism problem

Let G = Z_3 direct product Z_3 direct product Z_3 and let H be the subgroup of SL(3, Z_3) consisting of 1 a b the matrix H = { 0 1 c with a, b, c in Z_3 } 0 0 1 What is the order of G and H and are G and H isomorphic?

Permutation problem

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.

Permutation problem

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?

Group problem

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 problem

List all of the elements of order 15 in Z_600. - I think the elements are 40*1, 40*2, 40*4, 40*6, 40*7, 40*8, 40*10, 40*11, 40*13

Cyclic group problem

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

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 ...continues

Abelian groups

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.

Arc, Angles, Measures of a Circle

Using the diagram attached: 11. What is the measure of arc AC? 12. If ABC is a 30-60-90 triangle, with angle ACB at 30 degrees, and line segment AC is the diameter of the circle, then if the length of line segment AB is 4, what is the radius of the circle? 13. Working with the information from 12 from here to #16, what is ...continues

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.