# Generators of Groups

• Oct 8th 2010, 10:42 PM
jzellt
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.
• Oct 9th 2010, 02:47 AM
Swlabr
Quote:

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, $\displaystyle <5> = \{5, 0\}$ as $\displaystyle 5+5=0$.

Quote:

Originally Posted by jzellt
b) Show that Z10 is generated by 2 and 5

$\displaystyle Z_{10} = <1>$. That is to say, it is generated by the element 1. So, if you can prove that $\displaystyle 1 \in <2, 5>$ then you are done. You can prove that $\displaystyle 1 \in <2, 5>$ by noting that 2 and 5 are coprime, so...

Quote:

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...