Is the Direct Product of Groups Associative and Have an Identity Element?

[joelkato1605]

Homework Statement

Prove G X H is a group.

Relevant Equations

N/A

So I that I need to prove the axioms: associativity, existence of the identity element, and existence of the right inverse.

For associativity I know that the binary operations of G and H have to already be associative, and the elements of G X H are made up of these binary operations, so therefore it is associative?

For identity element let G have identity element e, and H have identity element f, can the identity element of G X H have the set (e,f), so that if (x,y) are included in G X H then (e,f) * (x,y) = (x,y) = (x,y) * (e,f)?

For the right inverse I am stuck, I don't know how to find it without knowing what the binary operations actually do.

 

[FactChecker]

Remember that G and H are groups, so you know that they each have an identity element and inverses for their respective operations. That should be enough. You don't need to know anything more about them.

 

[Office_Shredder]

For associativity I know that the binary operations of G and H have to already be associative, and the elements of G X H are made up of these binary operations, so therefore it is associative?

I think you might know what you're trying to say, but the words here are wrong. The elements of GxH are not made up of the binary operations on G and H, they are made from elements of G and H.

I think it is a good exercise to actually write down (ab)c and show how it transforms into a(bc) step by step.