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Math Help - Group isomorphism

  1. #1
    Member javax's Avatar
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    Group isomorphism

    Hello. Need some fast help on this:

    Let G, \star)\rightarrow(H, \triangle) " alt="\phi G, \star)\rightarrow(H, \triangle) " /> be an isomorphism between these two groups. If a' \in G is the inverse element of a \in G, then prove that \phi(a')\in H is the inverse element of \phi(a) \in H.

    I tried like this:
    Since \phi is isomorphism, that means it is a one to one map thus
    \phi(a)=\phi(b) only when a = b and since \phi is isomorphism, \phi(a') must be the inverse of \phi(a).

    Do you think my prove is enough, if not please elaborate.

    Thanks for your time.
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  2. #2
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    Quote Originally Posted by javax View Post
    Hello. Need some fast help on this:

    Let G, \star)\rightarrow(H, \triangle) " alt="\phi G, \star)\rightarrow(H, \triangle) " /> be an isomorphism between these two groups. If a' \in G is the inverse element of a \in G, then prove that \phi(a')\in H is the inverse element of \phi(a) \in H.

    I tried like this:
    Since \phi is isomorphism, that means it is a one to one map thus
    \phi(a)=\phi(b) only when a = b and since \phi is isomorphism, \phi(a') must be the inverse of \phi(a).


    I can't see any explanation or justification here...in fact, the argument is true for ANY group homomorphism, isomorphism or not, and the proof is boringly simple:

    \phi(a)\phi(a')=\phi(aa')=\phi(e_G)=e_H

    Tonio

    Do you think my prove is enough, if not please elaborate.

    Thanks for your time.
    .
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