Orders of Groups

  1. Hi i have completed the answer to this question. Just need your verification on whether it's completely correct or not:

    Question:
    If G is a group and xEG we define the order ord(x) by:
    ord(x) = min{[itex]r \geq 1: x^r = 1[/itex]}

    If [itex]\theta[/itex]: G --> H is an injective group homomorphism show that, for each xEG, ord([itex]\theta(x)[/itex]) = ord(x)

    My answer: Please verify
    If [itex]\theta(x)[/itex] = {[itex]x^r: r \epsilon Z[/itex]} then ord([itex]\theta(x)[/itex]) = ord(x).

    For any integer r, we have x^r = e (or 1) if and only if ord(x) divides r.

    In general the order of any subgroup of G divides the order of G. If H is a subgroup of G then "ord (G) / ord(H) = [G:H]" where [G:H] is an index of H in G, an integer.
    So order for any xEG divides order of the group. So ord([itex]\theta(x)[/itex]) = ord(x)


    any suggestions or changes please? thnx :)
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,306
    Staff Emeritus
    Science Advisor

    This makes no sense. [itex]\theta(x)[/itex] is a single member of H, not a set of members of G.

    An "index of H in G"? It is not said here that H has to be a subset of G!

    Seems to me you could just use the fact that, for any injective homomorphism, [itex]\theta[/itex], [itex]\theta(x^r)= [\theta(x)]^r[/itex] and [itex]\theta(1_G)= 1_H[/itex].
     
  4. i believe we have to show 2 things:

    i) [itex](\theta(x))^a = e'[/itex]
    ii) [itex]0 < b < a \implies (\theta(x))^b \neq e'[/itex].

    ok so basically:

    If ord(x)=a then [itex]\left[ {\phi (x)} \right]^a = \left[ {\phi (x^a )} \right] = \phi (e) = e'[/itex].
    Now suppose that [itex]ord\left[ {\phi (x)} \right] = b < a[/itex].
    Then
    [itex]\left[ {\phi (x)} \right]^b = e' = \left[ {\phi (x)} \right]^a [/itex]
    [itex]\phi (x^b ) = \phi (x^a )[/itex]
    [itex]x^b = x^a[/itex] (injective)
    [itex]x^{a - b} = e[/itex]

    there seems to be a contradiction where if [itex]x^{a} = x^{b}[/itex], then [itex]\left[ {\phi (x)} \right]^b = e'[/itex] which is not what statement (ii) says.
    am i correct in this assumption? any ideas on how to deal with this?
     
  5. When you have [itex]x^{a-b} = e[/itex], then a-b is positive since you assumed b<a. But this contradicts the definition of order. Done.
     
    Last edited: Jan 16, 2008
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