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 for full problem description.

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 the measure of arc AB? 14. What is the measure of arc BC? 15. Are arcs ABC and AC equal? 16. What is the measure of chord BC? 17. If the measure of arc AB is 75 degrees, what is the measure of arc BC? 18. If the measure of arc AB is 80 degrees, what is the measure of angle CAB? Hint: Remember that all the angles in a triangle add up to 180 degrees. 19. If I cre... click for more

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.