301.9F Internal Direct Product of Groups

[@essadababneh5871]

What an amazing educator. Please never delete your tutorials and lectures. They are perfect.

[@PunmasterSTP]

I was kind of confused about the distinction between internal and external direct products, but I think I understand now. External direct products are when you break out each subgroup into its own coordinate of an ordered pair, and internal direct products are where you just multiply elements of them together using the original group operation. Does that sound alright? Thanks!

[@heisenbergmuzik4948]

How to prove Z5 *Z7 (direct product) isomorphic to Z35.?? Would it be enough to show |(1,1)| = 35 ~ generator to generator. Plz help.

[@MatthewSalomone]

Yes! If you can show that the order of the group Z5 x Z7 is 35, and that the group has an element of order 35, as you say, then you have shown the group is cyclic. (And the generator-to-generator argument you mention would then establish isomorphism to Z35 in particular.)

[@heisenbergmuzik4948]

@@MatthewSalomone thank you very much sir.