Let $G$ be a group, and the order of the group $G$, $|G|=r$. For any $x in G$, prove that $x^{-r}=e$ (where...











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Let $G$ be a finite group, and the order of the group $G$, $|G|=r$.



For any $x in G$, prove that $x^{-r}=e$ (where $e$ is a neutral in $G$).



Basically, I haven't tried anything except trying to manipulate the expression $aa^{-1}=e$ into what I'm trying to prove because I really have no idea what to do.



I would appreciate any hint, advice or etc. because I'm new in group theory and the concepts are still hard to understand.



Also, the next part of this task is this:



$G$ is a group such that $|G|=195$. Is there an element $g neq e$ such that $g^{77}=e$?



Thanks in advance!










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  • 1




    For your first question, it might help to note that $(x^{-r}) = (x^{-1})^r = (x^r)^{-1}$. Either of those latter equalities, with the definition of order of a group and the fact that each $x in G$ has an inverse, should give you a push forward. I'm not 100% sure how to approach your second question though (don't have the best foundation in this topic), so I'll refrain from commenting on that.
    – Eevee Trainer
    yesterday






  • 1




    You need Lagrange's theorem.
    – the_fox
    yesterday










  • I'm not quite sure how to use that @the_fox
    – mathbbandstuff
    yesterday








  • 1




    For the second question, it suffices to observe that none of the factors of 77 divide the order of the group, which is necessary per Lagrange's theorem for a nonzero element to equal 1 when raised to the 77th power.
    – Monstrous Moonshiner
    yesterday















up vote
1
down vote

favorite












Let $G$ be a finite group, and the order of the group $G$, $|G|=r$.



For any $x in G$, prove that $x^{-r}=e$ (where $e$ is a neutral in $G$).



Basically, I haven't tried anything except trying to manipulate the expression $aa^{-1}=e$ into what I'm trying to prove because I really have no idea what to do.



I would appreciate any hint, advice or etc. because I'm new in group theory and the concepts are still hard to understand.



Also, the next part of this task is this:



$G$ is a group such that $|G|=195$. Is there an element $g neq e$ such that $g^{77}=e$?



Thanks in advance!










share|cite|improve this question




















  • 1




    For your first question, it might help to note that $(x^{-r}) = (x^{-1})^r = (x^r)^{-1}$. Either of those latter equalities, with the definition of order of a group and the fact that each $x in G$ has an inverse, should give you a push forward. I'm not 100% sure how to approach your second question though (don't have the best foundation in this topic), so I'll refrain from commenting on that.
    – Eevee Trainer
    yesterday






  • 1




    You need Lagrange's theorem.
    – the_fox
    yesterday










  • I'm not quite sure how to use that @the_fox
    – mathbbandstuff
    yesterday








  • 1




    For the second question, it suffices to observe that none of the factors of 77 divide the order of the group, which is necessary per Lagrange's theorem for a nonzero element to equal 1 when raised to the 77th power.
    – Monstrous Moonshiner
    yesterday













up vote
1
down vote

favorite









up vote
1
down vote

favorite











Let $G$ be a finite group, and the order of the group $G$, $|G|=r$.



For any $x in G$, prove that $x^{-r}=e$ (where $e$ is a neutral in $G$).



Basically, I haven't tried anything except trying to manipulate the expression $aa^{-1}=e$ into what I'm trying to prove because I really have no idea what to do.



I would appreciate any hint, advice or etc. because I'm new in group theory and the concepts are still hard to understand.



Also, the next part of this task is this:



$G$ is a group such that $|G|=195$. Is there an element $g neq e$ such that $g^{77}=e$?



Thanks in advance!










share|cite|improve this question















Let $G$ be a finite group, and the order of the group $G$, $|G|=r$.



For any $x in G$, prove that $x^{-r}=e$ (where $e$ is a neutral in $G$).



Basically, I haven't tried anything except trying to manipulate the expression $aa^{-1}=e$ into what I'm trying to prove because I really have no idea what to do.



I would appreciate any hint, advice or etc. because I'm new in group theory and the concepts are still hard to understand.



Also, the next part of this task is this:



$G$ is a group such that $|G|=195$. Is there an element $g neq e$ such that $g^{77}=e$?



Thanks in advance!







abstract-algebra group-theory finite-groups






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share|cite|improve this question













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share|cite|improve this question








edited yesterday

























asked yesterday









mathbbandstuff

242110




242110








  • 1




    For your first question, it might help to note that $(x^{-r}) = (x^{-1})^r = (x^r)^{-1}$. Either of those latter equalities, with the definition of order of a group and the fact that each $x in G$ has an inverse, should give you a push forward. I'm not 100% sure how to approach your second question though (don't have the best foundation in this topic), so I'll refrain from commenting on that.
    – Eevee Trainer
    yesterday






  • 1




    You need Lagrange's theorem.
    – the_fox
    yesterday










  • I'm not quite sure how to use that @the_fox
    – mathbbandstuff
    yesterday








  • 1




    For the second question, it suffices to observe that none of the factors of 77 divide the order of the group, which is necessary per Lagrange's theorem for a nonzero element to equal 1 when raised to the 77th power.
    – Monstrous Moonshiner
    yesterday














  • 1




    For your first question, it might help to note that $(x^{-r}) = (x^{-1})^r = (x^r)^{-1}$. Either of those latter equalities, with the definition of order of a group and the fact that each $x in G$ has an inverse, should give you a push forward. I'm not 100% sure how to approach your second question though (don't have the best foundation in this topic), so I'll refrain from commenting on that.
    – Eevee Trainer
    yesterday






  • 1




    You need Lagrange's theorem.
    – the_fox
    yesterday










  • I'm not quite sure how to use that @the_fox
    – mathbbandstuff
    yesterday








  • 1




    For the second question, it suffices to observe that none of the factors of 77 divide the order of the group, which is necessary per Lagrange's theorem for a nonzero element to equal 1 when raised to the 77th power.
    – Monstrous Moonshiner
    yesterday








1




1




For your first question, it might help to note that $(x^{-r}) = (x^{-1})^r = (x^r)^{-1}$. Either of those latter equalities, with the definition of order of a group and the fact that each $x in G$ has an inverse, should give you a push forward. I'm not 100% sure how to approach your second question though (don't have the best foundation in this topic), so I'll refrain from commenting on that.
– Eevee Trainer
yesterday




For your first question, it might help to note that $(x^{-r}) = (x^{-1})^r = (x^r)^{-1}$. Either of those latter equalities, with the definition of order of a group and the fact that each $x in G$ has an inverse, should give you a push forward. I'm not 100% sure how to approach your second question though (don't have the best foundation in this topic), so I'll refrain from commenting on that.
– Eevee Trainer
yesterday




1




1




You need Lagrange's theorem.
– the_fox
yesterday




You need Lagrange's theorem.
– the_fox
yesterday












I'm not quite sure how to use that @the_fox
– mathbbandstuff
yesterday






I'm not quite sure how to use that @the_fox
– mathbbandstuff
yesterday






1




1




For the second question, it suffices to observe that none of the factors of 77 divide the order of the group, which is necessary per Lagrange's theorem for a nonzero element to equal 1 when raised to the 77th power.
– Monstrous Moonshiner
yesterday




For the second question, it suffices to observe that none of the factors of 77 divide the order of the group, which is necessary per Lagrange's theorem for a nonzero element to equal 1 when raised to the 77th power.
– Monstrous Moonshiner
yesterday










2 Answers
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1
down vote



accepted










Suppose $o(x)=d$. This is the same as saying that $o(langle x rangle)=d$. By Lagrange's Theorem (which I assume you know), $d$ divides $r$, so $r/d$ is an integer, call it $m$. Then $x^r = (x^d)^m=e^m=e$.



For the second question: if there is such a $g$, then since $g^{77}=e$, the order of $g$ must be a divisor of $77$ (do you see why?). But, again by Lagrange, the order of $g$ must also divide $195 = |G|$, thus it must also divide $gcd(195,77)=1$. The only element of order $1$ in any group is $e$, which is a contradiction to the assumption that $g neq e$.






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    If $g^{77}=1$, then $(g^{77})^2=g^{154}=1$ and $(g^{77})^3=g^{231}=g^{231(mod 195)}=g^{36}=1$.



    Notice $195=3*5*13$ and $77=7*11$ so $lcm(77,195)=77*195$ which means $77*a neq 0(mod195)$ for $a<195$. So by applying the above argument, $(g^{77})^{38}=g=1$. So contradiction.






    share|cite|improve this answer





















    • Why $(g^{77})^{38}=g$? Sorry if I don't see something basic. Also, do you know how to solve the other problem?
      – mathbbandstuff
      yesterday






    • 1




      Sorry, I used online calculator, because 77 and 38 are inverse of each other in (mod 195). So 77*38=1(mod 195).
      – mathnoob
      yesterday










    • Oh, didn't think of that :) Thanks!
      – mathbbandstuff
      yesterday













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    2 Answers
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    Suppose $o(x)=d$. This is the same as saying that $o(langle x rangle)=d$. By Lagrange's Theorem (which I assume you know), $d$ divides $r$, so $r/d$ is an integer, call it $m$. Then $x^r = (x^d)^m=e^m=e$.



    For the second question: if there is such a $g$, then since $g^{77}=e$, the order of $g$ must be a divisor of $77$ (do you see why?). But, again by Lagrange, the order of $g$ must also divide $195 = |G|$, thus it must also divide $gcd(195,77)=1$. The only element of order $1$ in any group is $e$, which is a contradiction to the assumption that $g neq e$.






    share|cite|improve this answer

























      up vote
      1
      down vote



      accepted










      Suppose $o(x)=d$. This is the same as saying that $o(langle x rangle)=d$. By Lagrange's Theorem (which I assume you know), $d$ divides $r$, so $r/d$ is an integer, call it $m$. Then $x^r = (x^d)^m=e^m=e$.



      For the second question: if there is such a $g$, then since $g^{77}=e$, the order of $g$ must be a divisor of $77$ (do you see why?). But, again by Lagrange, the order of $g$ must also divide $195 = |G|$, thus it must also divide $gcd(195,77)=1$. The only element of order $1$ in any group is $e$, which is a contradiction to the assumption that $g neq e$.






      share|cite|improve this answer























        up vote
        1
        down vote



        accepted







        up vote
        1
        down vote



        accepted






        Suppose $o(x)=d$. This is the same as saying that $o(langle x rangle)=d$. By Lagrange's Theorem (which I assume you know), $d$ divides $r$, so $r/d$ is an integer, call it $m$. Then $x^r = (x^d)^m=e^m=e$.



        For the second question: if there is such a $g$, then since $g^{77}=e$, the order of $g$ must be a divisor of $77$ (do you see why?). But, again by Lagrange, the order of $g$ must also divide $195 = |G|$, thus it must also divide $gcd(195,77)=1$. The only element of order $1$ in any group is $e$, which is a contradiction to the assumption that $g neq e$.






        share|cite|improve this answer












        Suppose $o(x)=d$. This is the same as saying that $o(langle x rangle)=d$. By Lagrange's Theorem (which I assume you know), $d$ divides $r$, so $r/d$ is an integer, call it $m$. Then $x^r = (x^d)^m=e^m=e$.



        For the second question: if there is such a $g$, then since $g^{77}=e$, the order of $g$ must be a divisor of $77$ (do you see why?). But, again by Lagrange, the order of $g$ must also divide $195 = |G|$, thus it must also divide $gcd(195,77)=1$. The only element of order $1$ in any group is $e$, which is a contradiction to the assumption that $g neq e$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered yesterday









        the_fox

        1,9271429




        1,9271429






















            up vote
            1
            down vote













            If $g^{77}=1$, then $(g^{77})^2=g^{154}=1$ and $(g^{77})^3=g^{231}=g^{231(mod 195)}=g^{36}=1$.



            Notice $195=3*5*13$ and $77=7*11$ so $lcm(77,195)=77*195$ which means $77*a neq 0(mod195)$ for $a<195$. So by applying the above argument, $(g^{77})^{38}=g=1$. So contradiction.






            share|cite|improve this answer





















            • Why $(g^{77})^{38}=g$? Sorry if I don't see something basic. Also, do you know how to solve the other problem?
              – mathbbandstuff
              yesterday






            • 1




              Sorry, I used online calculator, because 77 and 38 are inverse of each other in (mod 195). So 77*38=1(mod 195).
              – mathnoob
              yesterday










            • Oh, didn't think of that :) Thanks!
              – mathbbandstuff
              yesterday

















            up vote
            1
            down vote













            If $g^{77}=1$, then $(g^{77})^2=g^{154}=1$ and $(g^{77})^3=g^{231}=g^{231(mod 195)}=g^{36}=1$.



            Notice $195=3*5*13$ and $77=7*11$ so $lcm(77,195)=77*195$ which means $77*a neq 0(mod195)$ for $a<195$. So by applying the above argument, $(g^{77})^{38}=g=1$. So contradiction.






            share|cite|improve this answer





















            • Why $(g^{77})^{38}=g$? Sorry if I don't see something basic. Also, do you know how to solve the other problem?
              – mathbbandstuff
              yesterday






            • 1




              Sorry, I used online calculator, because 77 and 38 are inverse of each other in (mod 195). So 77*38=1(mod 195).
              – mathnoob
              yesterday










            • Oh, didn't think of that :) Thanks!
              – mathbbandstuff
              yesterday















            up vote
            1
            down vote










            up vote
            1
            down vote









            If $g^{77}=1$, then $(g^{77})^2=g^{154}=1$ and $(g^{77})^3=g^{231}=g^{231(mod 195)}=g^{36}=1$.



            Notice $195=3*5*13$ and $77=7*11$ so $lcm(77,195)=77*195$ which means $77*a neq 0(mod195)$ for $a<195$. So by applying the above argument, $(g^{77})^{38}=g=1$. So contradiction.






            share|cite|improve this answer












            If $g^{77}=1$, then $(g^{77})^2=g^{154}=1$ and $(g^{77})^3=g^{231}=g^{231(mod 195)}=g^{36}=1$.



            Notice $195=3*5*13$ and $77=7*11$ so $lcm(77,195)=77*195$ which means $77*a neq 0(mod195)$ for $a<195$. So by applying the above argument, $(g^{77})^{38}=g=1$. So contradiction.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered yesterday









            mathnoob

            63511




            63511












            • Why $(g^{77})^{38}=g$? Sorry if I don't see something basic. Also, do you know how to solve the other problem?
              – mathbbandstuff
              yesterday






            • 1




              Sorry, I used online calculator, because 77 and 38 are inverse of each other in (mod 195). So 77*38=1(mod 195).
              – mathnoob
              yesterday










            • Oh, didn't think of that :) Thanks!
              – mathbbandstuff
              yesterday




















            • Why $(g^{77})^{38}=g$? Sorry if I don't see something basic. Also, do you know how to solve the other problem?
              – mathbbandstuff
              yesterday






            • 1




              Sorry, I used online calculator, because 77 and 38 are inverse of each other in (mod 195). So 77*38=1(mod 195).
              – mathnoob
              yesterday










            • Oh, didn't think of that :) Thanks!
              – mathbbandstuff
              yesterday


















            Why $(g^{77})^{38}=g$? Sorry if I don't see something basic. Also, do you know how to solve the other problem?
            – mathbbandstuff
            yesterday




            Why $(g^{77})^{38}=g$? Sorry if I don't see something basic. Also, do you know how to solve the other problem?
            – mathbbandstuff
            yesterday




            1




            1




            Sorry, I used online calculator, because 77 and 38 are inverse of each other in (mod 195). So 77*38=1(mod 195).
            – mathnoob
            yesterday




            Sorry, I used online calculator, because 77 and 38 are inverse of each other in (mod 195). So 77*38=1(mod 195).
            – mathnoob
            yesterday












            Oh, didn't think of that :) Thanks!
            – mathbbandstuff
            yesterday






            Oh, didn't think of that :) Thanks!
            – mathbbandstuff
            yesterday




















             

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