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Math Help - Question from a test i had today about group proof

  1. #1
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    Post Question from a test i had today about group proof

    Okay here is the question and ill give the answer i put i dont know if its sufficient or if there was another way.

    Given G is a finite group whose only subgroups are {e} and G itself

    A)Show G is cyclic

    2 choices for G either G = {e} and G is cyclic. If not then there are more elements in G and since the only subgroups are the ones mentioned these elements must generate the entire group or else there would be another subgroup so there is a generator and G is cyclic.

    B)Show G is of prime order

    G is cyclic so G is isomorphic to Zn then we know G has subgroups generated by elements that divide order of G. There are only those 2 so order G is only divisible by 1 and itself so n is prime
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by ChrisBickle View Post
    Okay here is the question and ill give the answer i put i dont know if its sufficient or if there was another way.

    Given G is a finite group whose only subgroups are {e} and G itself

    A)Show G is cyclic
    If G=\{e\} we are done. Otherwise, let g\in G then clearly \left\langle g\right\rangle\leqslant G and since it's non-trivial it must be improper. Thus, \left\langle g\right\rangle =G


    B)Show G is of prime order

    G is cyclic so G is isomorphic to Zn then we know G has subgroups generated by elements that divide order of G. There are only those 2 so order G is only divisible by 1 and itself so n is prime
    That's good!
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