Automorphism of a non-abelian finite group sending 3/4 of elements to inverses

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Taken From Topic in ALgebra herstein, Page No .71, question No.12



can you find an example of a finite group which is
non-abelian and which has an automorphism which maps exactly
three-quarters of the elements of G onto their inverses?



My attempts : my professor said me That take $mathbbQ_8$(Quaternion group).



i take $T= beginbmatrix 1 &-1&i&-i&j&-j&k&-k \ 1& -1&-i&i&-j&j&k&-k endbmatrix$



Now How can i show that $ T $ is an auto morphism of $mathbbQ$ which transfer exactly $frac34$ element of $mathbbQ$ into their inverse



Pliz help me,,,



thanks u




abstract-algebra




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  • 3




    That title is rather misleading.
    – Lord Shark the Unknown
    17 hours ago










  • okk@LordSharktheUnknown..then what title i have to put
    – stupid
    17 hours ago






  • 2




    "How can i show that $T$ is ..." $leftarrow$ what is $T$?
    – Théophile
    17 hours ago







  • 1




    Well, you can see that $T$ maps exactly $3/4$ of the elements onto their inverses. Now prove that $T$ is an automorphism.
    – Théophile
    17 hours ago






  • 1




    That certainly maps six elements to their inverses, but is it an automorphism? Could it be an inner automorphism say?
    – Lord Shark the Unknown
    17 hours ago














up vote
0
down vote

favorite












Taken From Topic in ALgebra herstein, Page No .71, question No.12



can you find an example of a finite group which is
non-abelian and which has an automorphism which maps exactly
three-quarters of the elements of G onto their inverses?



My attempts : my professor said me That take $mathbbQ_8$(Quaternion group).



i take $T= beginbmatrix 1 &-1&i&-i&j&-j&k&-k \ 1& -1&-i&i&-j&j&k&-k endbmatrix$



Now How can i show that $ T $ is an auto morphism of $mathbbQ$ which transfer exactly $frac34$ element of $mathbbQ$ into their inverse



Pliz help me,,,



thanks u




abstract-algebra




share|cite|improve this question

















  • 3




    That title is rather misleading.
    – Lord Shark the Unknown
    17 hours ago










  • okk@LordSharktheUnknown..then what title i have to put
    – stupid
    17 hours ago






  • 2




    "How can i show that $T$ is ..." $leftarrow$ what is $T$?
    – Théophile
    17 hours ago







  • 1




    Well, you can see that $T$ maps exactly $3/4$ of the elements onto their inverses. Now prove that $T$ is an automorphism.
    – Théophile
    17 hours ago






  • 1




    That certainly maps six elements to their inverses, but is it an automorphism? Could it be an inner automorphism say?
    – Lord Shark the Unknown
    17 hours ago












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Taken From Topic in ALgebra herstein, Page No .71, question No.12



can you find an example of a finite group which is
non-abelian and which has an automorphism which maps exactly
three-quarters of the elements of G onto their inverses?



My attempts : my professor said me That take $mathbbQ_8$(Quaternion group).



i take $T= beginbmatrix 1 &-1&i&-i&j&-j&k&-k \ 1& -1&-i&i&-j&j&k&-k endbmatrix$



Now How can i show that $ T $ is an auto morphism of $mathbbQ$ which transfer exactly $frac34$ element of $mathbbQ$ into their inverse



Pliz help me,,,



thanks u




abstract-algebra




share|cite|improve this question













Taken From Topic in ALgebra herstein, Page No .71, question No.12



can you find an example of a finite group which is
non-abelian and which has an automorphism which maps exactly
three-quarters of the elements of G onto their inverses?



My attempts : my professor said me That take $mathbbQ_8$(Quaternion group).



i take $T= beginbmatrix 1 &-1&i&-i&j&-j&k&-k \ 1& -1&-i&i&-j&j&k&-k endbmatrix$



Now How can i show that $ T $ is an auto morphism of $mathbbQ$ which transfer exactly $frac34$ element of $mathbbQ$ into their inverse



Pliz help me,,,



thanks u





abstract-algebra





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share|cite|improve this question








edited 17 hours ago
























asked 17 hours ago









stupid

4888




4888







  • 3




    That title is rather misleading.
    – Lord Shark the Unknown
    17 hours ago










  • okk@LordSharktheUnknown..then what title i have to put
    – stupid
    17 hours ago






  • 2




    "How can i show that $T$ is ..." $leftarrow$ what is $T$?
    – Théophile
    17 hours ago







  • 1




    Well, you can see that $T$ maps exactly $3/4$ of the elements onto their inverses. Now prove that $T$ is an automorphism.
    – Théophile
    17 hours ago






  • 1




    That certainly maps six elements to their inverses, but is it an automorphism? Could it be an inner automorphism say?
    – Lord Shark the Unknown
    17 hours ago












  • 3




    That title is rather misleading.
    – Lord Shark the Unknown
    17 hours ago










  • okk@LordSharktheUnknown..then what title i have to put
    – stupid
    17 hours ago






  • 2




    "How can i show that $T$ is ..." $leftarrow$ what is $T$?
    – Théophile
    17 hours ago







  • 1




    Well, you can see that $T$ maps exactly $3/4$ of the elements onto their inverses. Now prove that $T$ is an automorphism.
    – Théophile
    17 hours ago






  • 1




    That certainly maps six elements to their inverses, but is it an automorphism? Could it be an inner automorphism say?
    – Lord Shark the Unknown
    17 hours ago







3




3




That title is rather misleading.
– Lord Shark the Unknown
17 hours ago




That title is rather misleading.
– Lord Shark the Unknown
17 hours ago












okk@LordSharktheUnknown..then what title i have to put
– stupid
17 hours ago




okk@LordSharktheUnknown..then what title i have to put
– stupid
17 hours ago




2




2




"How can i show that $T$ is ..." $leftarrow$ what is $T$?
– Théophile
17 hours ago





"How can i show that $T$ is ..." $leftarrow$ what is $T$?
– Théophile
17 hours ago





1




1




Well, you can see that $T$ maps exactly $3/4$ of the elements onto their inverses. Now prove that $T$ is an automorphism.
– Théophile
17 hours ago




Well, you can see that $T$ maps exactly $3/4$ of the elements onto their inverses. Now prove that $T$ is an automorphism.
– Théophile
17 hours ago




1




1




That certainly maps six elements to their inverses, but is it an automorphism? Could it be an inner automorphism say?
– Lord Shark the Unknown
17 hours ago




That certainly maps six elements to their inverses, but is it an automorphism? Could it be an inner automorphism say?
– Lord Shark the Unknown
17 hours ago










1 Answer
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An automorphism is an isomorphism $T:Gto G$. I.e. An isomorphism to itself. You explicitly defined a permutation of elements so it is certainly a bijection.



It therefore suffices to show that $T$ is a homomorphism. This is rather awkward if you want to check it directly. (Because you have a lot of combinations to check.)



It is somewhat easier if you define what happens to the generating elements and extend the map to a homomorphism and check that they satisfy the given relations.



I.e $Q_8=langle i,j,k: i^4=j^4=k^4=1, ij=k, jk=i, ki=jrangle $ And define $T(I), T(j), T(k)$ and show they satisfy the given relations.






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    up vote
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    An automorphism is an isomorphism $T:Gto G$. I.e. An isomorphism to itself. You explicitly defined a permutation of elements so it is certainly a bijection.



    It therefore suffices to show that $T$ is a homomorphism. This is rather awkward if you want to check it directly. (Because you have a lot of combinations to check.)



    It is somewhat easier if you define what happens to the generating elements and extend the map to a homomorphism and check that they satisfy the given relations.



    I.e $Q_8=langle i,j,k: i^4=j^4=k^4=1, ij=k, jk=i, ki=jrangle $ And define $T(I), T(j), T(k)$ and show they satisfy the given relations.






    share|cite|improve this answer

























      up vote
      1
      down vote













      An automorphism is an isomorphism $T:Gto G$. I.e. An isomorphism to itself. You explicitly defined a permutation of elements so it is certainly a bijection.



      It therefore suffices to show that $T$ is a homomorphism. This is rather awkward if you want to check it directly. (Because you have a lot of combinations to check.)



      It is somewhat easier if you define what happens to the generating elements and extend the map to a homomorphism and check that they satisfy the given relations.



      I.e $Q_8=langle i,j,k: i^4=j^4=k^4=1, ij=k, jk=i, ki=jrangle $ And define $T(I), T(j), T(k)$ and show they satisfy the given relations.






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        An automorphism is an isomorphism $T:Gto G$. I.e. An isomorphism to itself. You explicitly defined a permutation of elements so it is certainly a bijection.



        It therefore suffices to show that $T$ is a homomorphism. This is rather awkward if you want to check it directly. (Because you have a lot of combinations to check.)



        It is somewhat easier if you define what happens to the generating elements and extend the map to a homomorphism and check that they satisfy the given relations.



        I.e $Q_8=langle i,j,k: i^4=j^4=k^4=1, ij=k, jk=i, ki=jrangle $ And define $T(I), T(j), T(k)$ and show they satisfy the given relations.






        share|cite|improve this answer













        An automorphism is an isomorphism $T:Gto G$. I.e. An isomorphism to itself. You explicitly defined a permutation of elements so it is certainly a bijection.



        It therefore suffices to show that $T$ is a homomorphism. This is rather awkward if you want to check it directly. (Because you have a lot of combinations to check.)



        It is somewhat easier if you define what happens to the generating elements and extend the map to a homomorphism and check that they satisfy the given relations.



        I.e $Q_8=langle i,j,k: i^4=j^4=k^4=1, ij=k, jk=i, ki=jrangle $ And define $T(I), T(j), T(k)$ and show they satisfy the given relations.







        share|cite|improve this answer













        share|cite|improve this answer



        share|cite|improve this answer











        answered 15 hours ago









        daruma

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