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Group thoery

  1. Sep 19, 2013 #1
    1. The problem statement, all variables and given/known data

    Show that any group of order 4 or less is abelian

    2. The attempt at a solution

    I came across this hint.

    Since its of order 4 we have {e,a,b,c}, where e = identity.

    The elements a, b, c must have order 2 or 4. There are two possibilities.

    1. a, b, c all have order 2.
    2. At least one of them, say a, has order 4.


    Why does the elements a, b, c have order 2 or 4? And why does at least one of them have order 4?
     
  2. jcsd
  3. Sep 19, 2013 #2
    Order is the exponent you have to raise them to obtain the identity?
    If the group is of order 4 you can suppose that you have at least one element with order of 3 and arrive a contradiction.
     
  4. Sep 19, 2013 #3
    Are you asking me a question or is that the answer to my question? lol
     
  5. Sep 19, 2013 #4

    CAF123

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    Gold Member

    The order of elements of the group have to divide the order of the group (Corollary of Lagrange) This does not tell us for sure that a group of order 4 has elements of order 2, only that 2 and 4 are the only possible orders. The element of order 1 is reserved as the identity element.
     
  6. Sep 19, 2013 #5
    Sorry, I didn't meant to confuse you, I was asking if the order of an element was the minimum exponent you need to raise it to obtain the identity. In other words, I was assuming that was the property "order" in my answer.
     
  7. Sep 19, 2013 #6
    Thank you guys!
     
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