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Thread: Cyclic Group

  1. #1
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    Cyclic Group

    Hello i was wondering if anybody could help me with the following question:


    Let (G,) be a group. Given a group action s : G → S4 having the property thats(f) = (1 2) and s(g) = (3 4) for certain elements f,g ∈ G. Can (G,) be a cyclicgroup?
    (Recall that a group (G,) is called cyclic if there exists γ ∈ G such thathγi, the subgroup generated by γ, is equal to G.)

    As I am completely lost.

    I think that the argument is something like there doesn't exist any number a such that (12)^a = (34), but I am not sure.

    King Regards
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  2. #2
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    Re: Cyclic Group

    As is standard, I assume you mean that s is a homomorphism from G into $S_4$. Can G be cyclic? No. If G were cyclic, then G/Ker(s) would be cyclic and would then contain two different elements of order 2 -- it is "easy" to see this is impossible in a cyclic group.
    Thanks from Krisly and topsquark
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  3. #3
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    Re: Cyclic Group

    Thank you for the reply. I am still having problem realizing when to use results regarding the quotient group.
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