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Math Help - Cyclic Subgroups of a finite group G

  1. #1
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    Cyclic Subgroups of a finite group G

    How can I show all elements of a cyclic subgroup are distinct?
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  2. #2
    Member ModusPonens's Avatar
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    Re: Cyclic Subgroups of a finite group G

    All elements of a finite cyclic group are of the form a^m, where a^n=e. Imagine that there were two elements a^m and a^k such that k is different from m, but a^m=a^k. Let m=na_1+r_1 and k=na_2+r_2 where the r's are the rest of the division by n and thus less than n. Lets assume, without loss of generality, that r_1 >r_2. Then assume a^m=a^{r_1} is equal to a^{r_2}=a^k and we'll get a contradiction. a^{r_1}=a^{r_2} implies a^{r_1-r_2}=e and thus there would be a number r_1-r_2<n such that a^{r_1-r_2}=e, which is a contradiction since n is the smallest natural number for which a^n=e.

    Now do the infinite cyclic one.
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