Cyclic Subgroups in Symmetric and Cyclic Groups

[chibulls59]

Suppose K= < x > is a cyclic group with 2 elements and H= S₃ is symmetric group with 6 elements. Find all different cyclic subgroups of G= H x K.

Now since K is generated by x with 2 elements, I have K= {1,x} and H= {1, (12), (13), (23), (123), (132)}

What I am confused about is finding cyclic subgroups of H x K. Am I supposed to be checking each element of H x K and seeing if it can generate the whole group?

 

[Dick]

All elements of the group generate a cyclic group. Some of them generate the same cyclic group. You are just supposed to figure out how many there are. The whole group isn't cyclic.