Direct Product of two Groups

  • #1

Main Question or Discussion Point

Suppose you had the following:

(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

ai , bi[tex] \in A,B
[/tex]

[(a1,b1)(a2,b2)](a3,b3)=(a1,b1)[(a2,b2)(a3,b3)]

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-11b-11 and so on until i get something like b3=b3

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.
 

Answers and Replies

  • #2
tiny-tim
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Hi chaotixmonjuish! :smile:
[(a1,b1)(a2,b2)](a3,b3)=(a1,b1)[(a2,b2)(a3,b3)]

By the way this question comes from Dummit and Foote, it says that the proof of this is left as a straightforward excercise.
Yup, it is straightforward …

i] what is [(a1,b1)(a2,b2)]?

ii] so what is [(a1,b1)(a2,b2)](a3,b3)?

:wink:
 
  • #3
Your hint kind of confused me.
 
  • #4
After multiplying everything

I'm changing the triangle to *'

(a1*a2,b1*'b2)(a3,b3)=(a1,b1)(a2*a3,b2*'b3)

Then once more:

(a1*a2*a3,b1*'b2*'b3)=
(a1*a2*a3,b1*'b2*'b3)

Since we know (A,*) and (B,*') is a group, we know there exists identity elements ai^1 and bi-1. We can use those identities to get all the a's equal to each other or the b's equal. That's shows associativity.
 
  • #5
tiny-tim
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hmm … a bit complicated …

try starting with [(a1,b1)(a2,b2)](a3,b3) = ([a1a2]a3,[b1b2]b3) :wink:
 
  • #6
Outside of that little thing, am I on the right track?
 
  • #7
tiny-tim
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hmm … I got very confused by your …
Since we know (A,*) and (B,*') is a group, we know there exists identity elements ai^1 and bi-1. We can use those identities to get all the a's equal to each other or the b's equal. That's shows associativity.
… which I still don't see the need for. :confused:

on looking back, your method seems to be the same as mine (with the first line left out)

This really is a "straightforward exercise" (as your book calls it), but you seem to have a knack for making it complicated.

Just start with my line, use the associative law on each side of the comma separately, then separate out (a1,b1), and you're done :smile:
 

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