Abstract Algebra direct products and cyclic groups

[lisagayle2008]

I can't figure this out!

Let G and H be groups, with a in G and b in H.
If G x H is a cyclic group, prove G and H are both cyclic.

 

[daon]

Hint: Let e=(e1,e2) be the identity in GxH.

Note that Gx{e2} and {e1}xH are subgroups.

 

[DrMike]

If G x H is a cyclic group

In other words, G x H has an element that generates the whole group. Call it k = (g,h).

prove G and H are both cyclic.

That means, you want to find an element of G that generates the whole of G,a dn similar for H.

Note that G x H contains (g,e[sub:2qfdha2a]H[/sub:2qfdha2a]) for all g in G, and each of these can be written as some power of k......

 

[daon]

Its simpler than that. The fundamental theorem of cyclic groups states that every subgroup of a cyclic group is cyclic. G and H are isomorphic to Gx{eH} and {eG}xH resp.