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Math Help - finite group question

  1. #1
    Senior Member Danneedshelp's Avatar
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    finite group question

    Prove of disprove: there exists a finite group with an element of infinite order.


    I am not convinced this holds, because a finite group will have the form \{e,a_{1},a_{2},...,a_{n}\}, where e is the identity element. Since groups are closed under their operation, I would think each element would eventually have to be the identity for some n\in{\mathbb{N}}; namely, a_{i}^{n}=e for some n\in{\mathbb{N}}.

    Not sure though, we just introduced the concept of order yesterday. I looked ahead in the book and read a little about cyclic groups, but I don't think any knoweldge of that is needed for this.
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    Quote Originally Posted by Danneedshelp View Post
    Prove of disprove: there exists a finite group with an element of infinite order.


    I am not convinced this holds, because a finite group will have the form \{e,a_{1},a_{2},...,a_{n}\}, where e is the identity element. Since groups are closed under their operation, I would think each element would eventually have to be the identity for some n\in{\mathbb{N}}; namely, a_{i}^{n}=e for some n\in{\mathbb{N}}.

    Not sure though, we just introduced the concept of order yesterday. I looked ahead in the book and read a little about cyclic groups, but I don't think any knoweldge of that is needed for this.


    If a group G has an element g of infinite order, then by closedness of the operation we have that

    the set \{g,\,g^2,\,g^3\,\ldots,\,g^n,\,\ldots\} belongs to G for all n\in\mathbb{N} , and since the order of g is

    infinite this set is infinite, so the group cannot be finite.

    Interestingly, the other way around is true: there are infinite groups all of which elements have finite order.

    Tonio
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