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Math Help - If the order of a group is prime number then the group is cyclic?

  1. #1
    Senior Member x3bnm's Avatar
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    If the order of a group is prime number then the group is cyclic?

    Let G be a group with a prime number p of elements. If a \in G where a \neq e,
    then the order of a is some integer m \neq 1.

    But then the cyclic group \langle a \rangle has m elements.

    By Lagrange's theorem, m must be a factor of p.

    But p is a prime number, and therefore m = p.

    It follows that \langle a \rangle has p elements, and is therefore all of G!

    Conclusion:

    If G is a group with a prime number p of elements, then G is a cyclic group. Furthermore,
    any element a \neq e in G is a generator of G.



    The author said that:

    "But then the cyclic group \langle a \rangle has m elements."

    I understand that the cyclic group has m elements. No problem there. I understand this statement is true for the generator a.

    But why the group G has to be definitely cyclic at the end?


    Why the cyclic group \langle a \rangle = G?


    Can't there be situations where aaaa....a \notin G because we didn't say that \langle a \rangle is a subgroup of G?

    So how did he come to the conclusion that G is cyclic?
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  2. #2
    Senior Member x3bnm's Avatar
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    Re: If the order of a group is prime number then the group is cyclic?

    Don't worry I found the solution. The answer is on Pinter's Abstract Algebra book page-111:

    If G is any group and a \in G, it is easy to see that:
    .......

    c) the set of all the powers of a is a subgroup of G

    This subgroup is called the cyclic subgroup of G generated by a.


    So the answer to my problem in my last post:

    \text{For all element } a \in G, \langle a \rangle \text{ is a subgroup of } G

    That's it. That's all I wanted to know.
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  3. #3
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    Re: If the order of a group is prime number then the group is cyclic?

    to answer your earlier question as asked:

    no, for any element a of G, any power of a is ALSO in G, by closure.
    Thanks from x3bnm
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  4. #4
    Senior Member x3bnm's Avatar
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    Re: If the order of a group is prime number then the group is cyclic?

    Quote Originally Posted by Deveno View Post
    to answer your earlier question as asked:

    no, for any element a of G, any power of a is ALSO in G, by closure.
    Thanks Deveno.
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