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Math Help - groups

  1. #1
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    groups

    can someone help me with this questuon please its fundamnetal to the topic and preventing me understand the topic!!

    Thanks

    Edgar

    if G is a group and x is an element of G we define the order ord(x) of x by

    ord(x) = min {r > or equal to 1 : x^r=1}

    if f:G maps to His an injective group homomorphism show that for each x
    ord (f(x)) = ord (x)
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  2. #2
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    Quote Originally Posted by edgar davids View Post

    if G is a group and x is an element of G we define the order ord(x) of x by

    ord(x) = min {r > or equal to 1 : x^r=1}

    if f:G maps to His an injective group homomorphism show that for each x
    ord (f(x)) = ord (x)
    These are finite groups, right?
    Because if not, how do you know that x has order?
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  3. #3
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    yes there are finite groups
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  4. #4
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    Quote Originally Posted by edgar davids View Post
    yes there are finite groups

    Because, f is one-to-one.
    Thus, a injective function between two finite sets.
    Now, H\leq G thus, |H|\leq |G| (again they are finite).
    But you cannot have a injective map from f:X\to Y where they are non-empty and |X|>|Y|. Thus, the only possible choice is that H=G, that is the subgroup is the group itself. Now a injective map between the same finite non-empty set is also surjective (Pigeonhole principle). Thus, what we really have f:G\to G and it is an automorphism.
    Thus, by homomorphism properties,
    [f(x)]^n=f(x^n)=f(e)=e*
    (The reason why f(e)=e is because it is an automorphism).
    Thus, we know the order of f(x) is at most n. If it was less than n by * we have that x has smaller order which cannot be contrary to our assumption.
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