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Direct Product of Groups

  1. Jun 17, 2012 #1
    I was reading on wikipedia on direct product of groups because I wanted find out if every subgroup of [itex]G \times H[/itex] is realised as a direct product of subgroups of G and H. Apparently it is not, because the diagonal subgroup in [itex]G \times G [/itex] disproves this. I'm a little confused, because I thought the proof I wrote was correct
    for a subgroup write [itex]A \times B [/itex] where A is a subset of G, and B a subset of H. Can't you show A is a subgroup of G using [itex] (g,1) [/itex] and analogously with B? For example
    m,n in A then [itex] (m,1),(n,1) [/itex] are in [itex]A \times B [/itex]. Hence [itex] (mn,1) [/itex] is and therefore mn is in A?

    There must be something wrong? Is the property true for certain type of groups? But I didn't use anything about G and H.
  2. jcsd
  3. Jun 17, 2012 #2
    Looks like you have shown that the product of any 2 subgroup A and B, A x B is a subgroup of the product group G x H.

    You have not shown the opposite, that each subgroup of G x H can be written as a product A x B. Because that is not the case, as the counter example shows.
  4. Jun 17, 2012 #3

    Here is the gist of this stuff: you can't do this. It is not true that any subgroup of the direct product [itex]\,G\times H\,[/itex] can be realized as a subset of the corresponding cartesian product, just as it is not true that any subset of a cartesian product is a cartesian product of subsets of the corresponding sets in the product...


  5. Jun 18, 2012 #4
    Thanks for your replies!

    I'm sorry I'm having a little trouble understanding. Isn't the cartesian product defined as the set of elements of the form (g,h). Then any subset is a set of this form as well, so it is another direct product? If it is, why aren't the summands subsets of their respective supersets?
  6. Jun 18, 2012 #5

    Very simple: take the set [itex]\,A:=\{1,2\}\,\text{ and its cartesian product}\,\,A\times A\,[/itex] , and look at the latter's diagonal subset [itex]\,D:=\{(1,1)\,,\,(2,2)\}\,[/itex].

    Well, try to represent [itex]\,D=X\times Y\,\,,\text{for some subsets}\,X,Y\subset A\,[/itex] (Hint: you can't).

    So, again, your claim in " Isn't the cartesian product defined as the set of elements of the

    form (g,h). Then any subset is a set of this form as well" is false.

  7. Jun 18, 2012 #6
    Ah, I forgot: the direct product includes all combinations of elements of the summands!
    I also kept thinking the diagonal subset was some kind of pathological example (with B=A), but of course this works for general sets.

    Thank you for explaining DonAntonio! :D
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