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Isomorphisms abelian help

  1. Aug 20, 2014 #1
    1. The problem statement, all variables and given/known data
    Let G be a group, A = G * G. In A, Let T = {(g, g)|g ε G}. Prove that T is isomorphic to G.

    2. Relevant equations

    3. The attempt at a solution
    A is abelian. Therefore, G * G is abelian. T is a subgroup of G.

    I am not sure if my above inferences are even correct. Can someone guide me as to the thought process on this please? Thank you.
  2. jcsd
  3. Aug 21, 2014 #2
    I don't see how you could have inferred any of those, or why you would need to.

    You are asked to show that ##G## and ##T## are isomorphic. There is an obvious candidate for an isomorphism, so you should just verify that it actually is one.
  4. Aug 21, 2014 #3


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    And for that matter, you haven't told us what G*G means.
  5. Aug 22, 2014 #4
    A = G * G, as in G cross G.
    I do not understand how T is directly isomorphic to G.
  6. Aug 22, 2014 #5


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    How'd you get A is abelian?

    If A is abelian, then obviously GxG is abelian since A=GxG.

    How can T be a subgroup of G when it's not a subset of G? It's a subset of A, right?

    How did you define the group multiplication for A?

  7. Aug 23, 2014 #6


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    There is nothing in the question to suggest this. If [itex]G[/itex] is not abelian then [itex]A[/itex] will not be abelian.

    Actually T is a subgroup of A.

    You need to find a bijection [itex]\phi : G \to T[/itex] such that [itex]\phi(g)\phi(h) = \phi(gh)[/itex] for every [itex]g \in G[/itex] and [itex]h \in G[/itex].

    It would be good to start by writing out the group operation of A, and see what happens when you restrict it to T.
  8. Aug 23, 2014 #7


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    What is the group operation on G*G? You have to know that before you can even talk about an isomorphism.
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