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Math Help - Order 4 groups are isomorphic to...

  1. #1
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    Order 4 groups are isomorphic to...

    Is it correct to say that any order 4 group is only isomorphic to either
    C4 or C2+C2 ?


    where C4 is the order 4 cyclic group and C2 the order 2 cyclic group
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  2. #2
    Ant
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    Re: Order 4 groups are isomorphic to...

    Yep, you're right
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  3. #3
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    Re: Order 4 groups are isomorphic to...

    if a group G of order 4 has an element of order 4, say x, then {x,x2,x3,x4 = e} must be all of G, so G = <x>, and is isomorphic to C4.

    if not, then all 3 non-identity elements must be of order 2 (the only other divisor (besides 1) of 4 left). pick any two of them, say a and b.

    now <a> and <b> are two subgroups of order 2. it is straightforward to show that G is isomorphic to <a> x <b> (which is obviously isomorphic to C2 x C2):

    e --> (e,e)
    a --> (a,e)
    b --> (e,b)
    ab --> (a,b) is the isomorphism.

    (proving ba = ab is the only "non-obvious thing" (which must be true if we have an isomorphism since (a,e)*(e,b) = (ae,eb) = (a,b) = (ea,be) = (e,b)*(a,e)). it is perhaps easiest to show that ba = e,a, or b leads to a contradiction).
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