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Is order = size ?

  1. Oct 31, 2012 #1
    Is "order" = "size"?

    I'm really confused... we refer to the "size" of a group as the "order"... But are they really equivalent? Can we prove that simply?

    Example:
    Say we have a group G and a subgroup H. Then take one of the cosets of H, say Hx... It has order = o(G)/o(G/H). But does this mean that in G/H, (Hx)^[o(G)/o(G/H)] = e = H?? Or do elements have different orders in different groups? I'm so confused.

    Oh, and this whole factor group thing... Does this only apply to normal subgroups?
     
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  3. Oct 31, 2012 #2

    HallsofIvy

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    Re: Is "order" = "size"?

    You seem to be talking about two different definitions of "order". The "order of a group" is the number of elements in the group- its "size". The "order of an element" of group G is the order of the subgroup of G generated by the element. In particular if the order of an element, x, is n then x^n= e, the identity of G. I don't know what you mean by a subgroup, (Hx), to a power.
     
  4. Oct 31, 2012 #3

    micromass

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    Re: Is "order" = "size"?

    No, they are not equivalent. One definition of order means the "size" of the group. The other definition of order is the order of an element g: it is the smallest positive integer n such that [itex]g^n=e[/itex].

    Why does it have that order? Sure, the coset has |H| elements. But the order is the smallest positive number n such that [itex](Hx)^n=H[/itex]. This is in general not o(G)/o(G/H).


    Yes.
     
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