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Thread: grouop of order 4

  1. #1
    Junior Member
    Mar 2009

    grouop of order 4

    1. Let G be a group of order 4. Prove that every element of G has order 1,2, or 4.

    2. Classify groups of order 4 by considering the following two cases:
    a. G contains an element of order 4
    b. Every element of G has order <4

    could anyone please help me to solve these problems?
    and what does it mean by 'Classify groups of order 4'?
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  2. #2
    ynj is offline
    Senior Member
    Jul 2009
    the group itself will have order 4,|e|=1
    for any element a, |a|=1 or 2 or 4
    if |a|=1 then a=e
    if there is an a such that |a|=4, than it would be a cyclic, thus isomorphic to Z(4), so it will also have subgroup of order 2.
    if such element does not exist, then 3 elements will have order 2, that is, isomorphic to Z(2)*Z(2).
    "classify"actually means classify the group up to isomorphism
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