Suppose you had the following:(adsbygoogle = window.adsbygoogle || []).push({});

(A,*) and (B,[tex]\nabla[/tex])

So to prove associativity, since I know that both A and B are groups, their direct product will be a group. Could I do the following

a_{i}, b_{i}[tex] \in A,B

[/tex]

[(a_{1},b_{1})(a_{2},b_{2})](a_{3},b_{3})=(a_{1},b_{1})[(a_{2},b_{2})(a_{3},b_{3})]

Since A and B are groups, I know they have distributing everything via the proper binary operations (I got kind of lazy at this point). Can I just multiply both sides by an a^{-1}_{1}b^{-1}_{1}and so on until i get something like b_{3}=b_{3}

I just want to make sure this is sort of hte process one uses to prove the direct product is a group. By the way this question comes from Dummit and Foote, it says that the proof of this is left as a straightforward excercise.

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# Direct Product of two Groups

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