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Math Help - How many elements of order 2 are there in this product of groups?

  1. #1
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    How many elements of order 2 are there in this product of groups?

    How many elements of order 2 are there in the following abelian group of order 16? :

    Z2 X Z2 X Z4

    where Z2 is the integers mod 2 and Z4 is the integers mod 4.

    How are these found?

    Thanks for any help


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  2. #2
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    Quote Originally Posted by Siknature View Post
    How many elements of order 2 are there in the following abelian group of order 16? :

    Z2 X Z2 X Z4

    where Z2 is the integers mod 2 and Z4 is the integers mod 4.

    How are these found?

    Thanks for any help


    In this case perhaps is easier to ask how many elements are NOT of order 2: first, note that every non-unit element is of order 2 or 4.

    Now, if we agree to write the elements of the group in the form (x,y,z)\,,\,\,x,y,=0,1\!\!\!\!\pmod 2\,,\,z=0,1,2,3\!\!\!\!\pmod 4, then an element has order 4 iff it has either 1\,\,\, or\,\,\, 3\!\!\!\!\pmod 4 in the last entry...can you now count them all?

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    can you now count them all?

    Tonio
    Thanks, yes, now i think that this means the answer must be 7 (8 if we include the identity)
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