Thread: Generators of Groups

a) List all the cyclic subgroups of <Z10, +> (Z10 = integers mod 10)

b) Show that Z10 is generated by 2 and 5

c) Show that Z2 x Z3 is a cyclic group

Any help is appreciated.... Thanks.

Originally Posted by jzellt

a) List all the cyclic subgroups of <Z10, +> (Z10 = integers mod 10)

Take every element and work out the subgroup generated by them. For example, as .

Originally Posted by jzellt

b) Show that Z10 is generated by 2 and 5

. That is to say, it is generated by the element 1. So, if you can prove that then you are done. You can prove that by noting that 2 and 5 are coprime, so...

Originally Posted by jzellt

c) Show that Z2 x Z3 is a cyclic group

You want to find a single element which generates this group. So, try a few...