1 Let G be a cyclic group of order 2016 How many elements of

1. Let G be a cyclic group of order 2016. How many elements of G have order:

1? ____

2? ____

3? ____

4? ____

5? ____

Modern Algebra (Abstract Algebra Theory and Applications)

Solution

Let G be a cyclic group of order 2016.

Elements of G having order:

1 = 1,2016 2016*1= 2016

2= 2,1008 1008*2= 2016

3= 3,672 672*3= 2016

4=4,504 504*4= 2016

5=This is not possible since it is a Prime number and not a odd or even number.

If we suppose G is a group of order 35, and let xGxG such that xe, from Lagrange\'s Theorem x will be of order 5, 7, or 35. If x is of order 35, then G is cyclic and thus has elements of order 5 and 7.