Does alternating group $A_5$ is a subgroup of $GL(4,mathbb Z)$?












3














Does alternating group $A_5$ is a subgroup of $GL(4,mathbb Z)$? I know $A_5$ has a 4-dimensional complex representation. But how to prove $A_5$ is a subgroup or not of $GL(4, mathbb Z)$?



In case $GL(4, mathbb Z)$ has a subgroup isomorphic to $A_5$, then I would like to know corresponding generators of that subgroup?



I want to know how many subgroups are there in $GL(4, Bbb Z)$ isomorphic to $A_5$. Is there a unique one?



Thank you so much in advance.






group-theory representation-theory







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    3














    Does alternating group $A_5$ is a subgroup of $GL(4,mathbb Z)$? I know $A_5$ has a 4-dimensional complex representation. But how to prove $A_5$ is a subgroup or not of $GL(4, mathbb Z)$?



    In case $GL(4, mathbb Z)$ has a subgroup isomorphic to $A_5$, then I would like to know corresponding generators of that subgroup?



    I want to know how many subgroups are there in $GL(4, Bbb Z)$ isomorphic to $A_5$. Is there a unique one?



    Thank you so much in advance.






    group-theory representation-theory







    share|cite|improve this question



























      3












      3








      3







      Does alternating group $A_5$ is a subgroup of $GL(4,mathbb Z)$? I know $A_5$ has a 4-dimensional complex representation. But how to prove $A_5$ is a subgroup or not of $GL(4, mathbb Z)$?



      In case $GL(4, mathbb Z)$ has a subgroup isomorphic to $A_5$, then I would like to know corresponding generators of that subgroup?



      I want to know how many subgroups are there in $GL(4, Bbb Z)$ isomorphic to $A_5$. Is there a unique one?



      Thank you so much in advance.






      group-theory representation-theory







      share|cite|improve this question















      Does alternating group $A_5$ is a subgroup of $GL(4,mathbb Z)$? I know $A_5$ has a 4-dimensional complex representation. But how to prove $A_5$ is a subgroup or not of $GL(4, mathbb Z)$?



      In case $GL(4, mathbb Z)$ has a subgroup isomorphic to $A_5$, then I would like to know corresponding generators of that subgroup?



      I want to know how many subgroups are there in $GL(4, Bbb Z)$ isomorphic to $A_5$. Is there a unique one?



      Thank you so much in advance.






      group-theory representation-theory




      group-theory representation-theory






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      share|cite|improve this question













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      edited 7 hours ago

























      asked 23 hours ago









      mathstudent

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          1 Answer
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          3














          For any $n$, $S_n$ embeds in $GL(n-1,Bbb Z)$. Therefore $A_n$ does too.



          Consider the vectors $v_1,ldots,v_n$ where $v_1,ldots,v_{n-1}$
          are the standard unit vectors in $Bbb Z^{n-1}$ and $v_n=-v_1-cdots-v_{n-1}$.
          By definition $v_1+cdots+v_n=0$. For each permutation $tauin S_n$, there's
          a unique endomorphism of $Bbb Z^{n-1}$ taking each $v_k$ to $v_{tau(k)}$
          (this works due to $v_1+cdots+v_n=0$). So $S_n$ embeds into $text{Aut}(
          Bbb Z^{n-1})
          =GL(n-1,Bbb Z)$          .



          In fact $A_n$ will embed in $SL(n-1,Bbb Z)$.






          share|cite|improve this answer





















          • Thanks. I would like to know corresponding generators of the subgroup of $GL(4, mathbb Z)$ that is isomorphic to $A_5$ if possible.
            – mathstudent
            22 hours ago












          • What are your favourite generators of $A_5$? @mathstudent
            – Lord Shark the Unknown
            22 hours ago










          • Generators of $A_5$: $<(1 2)(3 4), (1 2 3 4 5)>$. Thanks.
            – mathstudent
            22 hours ago












          • $(1,2,3,4,5)$ maps to $pmatrix{0&0&0&-1\1&0&0&-1\0&1&0&-1\0&0&1&-1}$ etc.
            – Lord Shark the Unknown
            22 hours ago










          • Does $(1,2,5,3,4)$ maps to $pmatrix{0&-1&0&1\1&-1&0&0\0&-1&1&0\0&-1&0&0}$ and $(1,2)(3,4)$ maps to $pmatrix{0&1&0&0\1&0&0&0\0&0&0&1\0&0&1&0}$ ?
            – mathstudent
            22 hours ago













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          1 Answer
          1






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          active

          oldest

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          active

          oldest

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          3














          For any $n$, $S_n$ embeds in $GL(n-1,Bbb Z)$. Therefore $A_n$ does too.



          Consider the vectors $v_1,ldots,v_n$ where $v_1,ldots,v_{n-1}$
          are the standard unit vectors in $Bbb Z^{n-1}$ and $v_n=-v_1-cdots-v_{n-1}$.
          By definition $v_1+cdots+v_n=0$. For each permutation $tauin S_n$, there's
          a unique endomorphism of $Bbb Z^{n-1}$ taking each $v_k$ to $v_{tau(k)}$
          (this works due to $v_1+cdots+v_n=0$). So $S_n$ embeds into $text{Aut}(
          Bbb Z^{n-1})
          =GL(n-1,Bbb Z)$          .



          In fact $A_n$ will embed in $SL(n-1,Bbb Z)$.






          share|cite|improve this answer





















          • Thanks. I would like to know corresponding generators of the subgroup of $GL(4, mathbb Z)$ that is isomorphic to $A_5$ if possible.
            – mathstudent
            22 hours ago












          • What are your favourite generators of $A_5$? @mathstudent
            – Lord Shark the Unknown
            22 hours ago










          • Generators of $A_5$: $<(1 2)(3 4), (1 2 3 4 5)>$. Thanks.
            – mathstudent
            22 hours ago












          • $(1,2,3,4,5)$ maps to $pmatrix{0&0&0&-1\1&0&0&-1\0&1&0&-1\0&0&1&-1}$ etc.
            – Lord Shark the Unknown
            22 hours ago










          • Does $(1,2,5,3,4)$ maps to $pmatrix{0&-1&0&1\1&-1&0&0\0&-1&1&0\0&-1&0&0}$ and $(1,2)(3,4)$ maps to $pmatrix{0&1&0&0\1&0&0&0\0&0&0&1\0&0&1&0}$ ?
            – mathstudent
            22 hours ago


















          3














          For any $n$, $S_n$ embeds in $GL(n-1,Bbb Z)$. Therefore $A_n$ does too.



          Consider the vectors $v_1,ldots,v_n$ where $v_1,ldots,v_{n-1}$
          are the standard unit vectors in $Bbb Z^{n-1}$ and $v_n=-v_1-cdots-v_{n-1}$.
          By definition $v_1+cdots+v_n=0$. For each permutation $tauin S_n$, there's
          a unique endomorphism of $Bbb Z^{n-1}$ taking each $v_k$ to $v_{tau(k)}$
          (this works due to $v_1+cdots+v_n=0$). So $S_n$ embeds into $text{Aut}(
          Bbb Z^{n-1})
          =GL(n-1,Bbb Z)$          .



          In fact $A_n$ will embed in $SL(n-1,Bbb Z)$.






          share|cite|improve this answer





















          • Thanks. I would like to know corresponding generators of the subgroup of $GL(4, mathbb Z)$ that is isomorphic to $A_5$ if possible.
            – mathstudent
            22 hours ago












          • What are your favourite generators of $A_5$? @mathstudent
            – Lord Shark the Unknown
            22 hours ago










          • Generators of $A_5$: $<(1 2)(3 4), (1 2 3 4 5)>$. Thanks.
            – mathstudent
            22 hours ago












          • $(1,2,3,4,5)$ maps to $pmatrix{0&0&0&-1\1&0&0&-1\0&1&0&-1\0&0&1&-1}$ etc.
            – Lord Shark the Unknown
            22 hours ago










          • Does $(1,2,5,3,4)$ maps to $pmatrix{0&-1&0&1\1&-1&0&0\0&-1&1&0\0&-1&0&0}$ and $(1,2)(3,4)$ maps to $pmatrix{0&1&0&0\1&0&0&0\0&0&0&1\0&0&1&0}$ ?
            – mathstudent
            22 hours ago
















          3












          3








          3






          For any $n$, $S_n$ embeds in $GL(n-1,Bbb Z)$. Therefore $A_n$ does too.



          Consider the vectors $v_1,ldots,v_n$ where $v_1,ldots,v_{n-1}$
          are the standard unit vectors in $Bbb Z^{n-1}$ and $v_n=-v_1-cdots-v_{n-1}$.
          By definition $v_1+cdots+v_n=0$. For each permutation $tauin S_n$, there's
          a unique endomorphism of $Bbb Z^{n-1}$ taking each $v_k$ to $v_{tau(k)}$
          (this works due to $v_1+cdots+v_n=0$). So $S_n$ embeds into $text{Aut}(
          Bbb Z^{n-1})
          =GL(n-1,Bbb Z)$          .



          In fact $A_n$ will embed in $SL(n-1,Bbb Z)$.






          share|cite|improve this answer












          For any $n$, $S_n$ embeds in $GL(n-1,Bbb Z)$. Therefore $A_n$ does too.



          Consider the vectors $v_1,ldots,v_n$ where $v_1,ldots,v_{n-1}$
          are the standard unit vectors in $Bbb Z^{n-1}$ and $v_n=-v_1-cdots-v_{n-1}$.
          By definition $v_1+cdots+v_n=0$. For each permutation $tauin S_n$, there's
          a unique endomorphism of $Bbb Z^{n-1}$ taking each $v_k$ to $v_{tau(k)}$
          (this works due to $v_1+cdots+v_n=0$). So $S_n$ embeds into $text{Aut}(
          Bbb Z^{n-1})
          =GL(n-1,Bbb Z)$          .



          In fact $A_n$ will embed in $SL(n-1,Bbb Z)$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 23 hours ago









          Lord Shark the Unknown

          101k958132




          101k958132












          • Thanks. I would like to know corresponding generators of the subgroup of $GL(4, mathbb Z)$ that is isomorphic to $A_5$ if possible.
            – mathstudent
            22 hours ago












          • What are your favourite generators of $A_5$? @mathstudent
            – Lord Shark the Unknown
            22 hours ago










          • Generators of $A_5$: $<(1 2)(3 4), (1 2 3 4 5)>$. Thanks.
            – mathstudent
            22 hours ago












          • $(1,2,3,4,5)$ maps to $pmatrix{0&0&0&-1\1&0&0&-1\0&1&0&-1\0&0&1&-1}$ etc.
            – Lord Shark the Unknown
            22 hours ago










          • Does $(1,2,5,3,4)$ maps to $pmatrix{0&-1&0&1\1&-1&0&0\0&-1&1&0\0&-1&0&0}$ and $(1,2)(3,4)$ maps to $pmatrix{0&1&0&0\1&0&0&0\0&0&0&1\0&0&1&0}$ ?
            – mathstudent
            22 hours ago




















          • Thanks. I would like to know corresponding generators of the subgroup of $GL(4, mathbb Z)$ that is isomorphic to $A_5$ if possible.
            – mathstudent
            22 hours ago












          • What are your favourite generators of $A_5$? @mathstudent
            – Lord Shark the Unknown
            22 hours ago










          • Generators of $A_5$: $<(1 2)(3 4), (1 2 3 4 5)>$. Thanks.
            – mathstudent
            22 hours ago












          • $(1,2,3,4,5)$ maps to $pmatrix{0&0&0&-1\1&0&0&-1\0&1&0&-1\0&0&1&-1}$ etc.
            – Lord Shark the Unknown
            22 hours ago










          • Does $(1,2,5,3,4)$ maps to $pmatrix{0&-1&0&1\1&-1&0&0\0&-1&1&0\0&-1&0&0}$ and $(1,2)(3,4)$ maps to $pmatrix{0&1&0&0\1&0&0&0\0&0&0&1\0&0&1&0}$ ?
            – mathstudent
            22 hours ago


















          Thanks. I would like to know corresponding generators of the subgroup of $GL(4, mathbb Z)$ that is isomorphic to $A_5$ if possible.
          – mathstudent
          22 hours ago






          Thanks. I would like to know corresponding generators of the subgroup of $GL(4, mathbb Z)$ that is isomorphic to $A_5$ if possible.
          – mathstudent
          22 hours ago














          What are your favourite generators of $A_5$? @mathstudent
          – Lord Shark the Unknown
          22 hours ago




          What are your favourite generators of $A_5$? @mathstudent
          – Lord Shark the Unknown
          22 hours ago












          Generators of $A_5$: $<(1 2)(3 4), (1 2 3 4 5)>$. Thanks.
          – mathstudent
          22 hours ago






          Generators of $A_5$: $<(1 2)(3 4), (1 2 3 4 5)>$. Thanks.
          – mathstudent
          22 hours ago














          $(1,2,3,4,5)$ maps to $pmatrix{0&0&0&-1\1&0&0&-1\0&1&0&-1\0&0&1&-1}$ etc.
          – Lord Shark the Unknown
          22 hours ago




          $(1,2,3,4,5)$ maps to $pmatrix{0&0&0&-1\1&0&0&-1\0&1&0&-1\0&0&1&-1}$ etc.
          – Lord Shark the Unknown
          22 hours ago












          Does $(1,2,5,3,4)$ maps to $pmatrix{0&-1&0&1\1&-1&0&0\0&-1&1&0\0&-1&0&0}$ and $(1,2)(3,4)$ maps to $pmatrix{0&1&0&0\1&0&0&0\0&0&0&1\0&0&1&0}$ ?
          – mathstudent
          22 hours ago






          Does $(1,2,5,3,4)$ maps to $pmatrix{0&-1&0&1\1&-1&0&0\0&-1&1&0\0&-1&0&0}$ and $(1,2)(3,4)$ maps to $pmatrix{0&1&0&0\1&0&0&0\0&0&0&1\0&0&1&0}$ ?
          – mathstudent
          22 hours ago




















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