Isomorphism of semidirect products [D&F]












5












$begingroup$


I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 184 Exercise 6):




Assume that $K$ is a cyclic group, $H$ is an arbitrary group and $varphi_1$ and $varphi_2$ are homomorphisms from $K$ into $text{Aut}(H)$ such that $varphi_1(K)$ and $varphi_2(K)$ are subgroups of $text{Aut}(H)$. If $K$ is infinite assume $varphi_1$ and $varphi_2$ are injective. Prove by constructing an explicit isomorphism that $H rtimes_{varphi_1} K cong H rtimes_{varphi_2} K$ (in particular, if the subgroups $varphi_1(K)$ and $varphi_2(K)$ are equal in $text{Aut}(H)$, then the resulting semidirect products are isomorphic). [Suppose $sigma varphi_1(K) sigma^{-1}=varphi_2(K)$ so that for some $a in mathbb{Z}$ we have $sigma varphi_1(k) sigma^{-1}=varphi_2(k)^a$ for all $k in K$. Show that the map $psi:H rtimes_{varphi_1} K to H rtimes_{varphi_2} K$ defined by $psi((h,k))=(sigma(h),k^a)$ is a homomorphism. Show $psi$ is bijective by constructing a 2-sided inverse.]






My attempt:
Let $K=langle x rangle$, and suppose $sigma in text{Aut}(H)$ conjugates $varphi_1(K)$ to $varphi_2(K)$, that is



$$sigma varphi_1(K) sigma^{-1}=varphi_2(K) tag{1}.$$



From this we find
$$sigma varphi_1(x) sigma^{-1}=varphi_2(x^a) tag{2} $$
for some $a in mathbb{Z}$. Since all elements of $K$ are powers $x^n$ of $x$, raising this equality to the $n$-th power gives
$$forall k in K:sigma varphi_1(k) sigma^{-1}=varphi_2(k^a)=varphi_2(k)^a tag{3}.$$



We now prove that the suggested $psi$ is a group homomorphism:



begin{equation}
begin{split}
psi((h_1,k_1)(h_2,k_2))&=psi(h_1 varphi_1(k_1)(h_2),k_1k_2)=(sigma(h_1 varphi_1(k_1)(h_2)),(k_1k_2)^a)\
&=(sigma(h_1)(sigma varphi_1(k_1))(h_2),k_1^a k_2^a)=(sigma(h_1)(varphi_2(k_1^a) sigma)(h_2),k_1^ak_2^a)\
&=(sigma(h_1),k_1^a)(sigma(h_2),k_2^a)=psi((h_1,k_1))psi((h_2,k_2))
end{split}
end{equation}



Where we have used the fact that $K$ is abelian, alongside with the homomorphism law for $sigma$ and equation $(3)$. We're left with showing that $psi$ has a 2-sided inverse, which will be done in two cases:






  • Assume $K=langle x rangle$ is infinite cyclic. Property $(3)$ gives
    $$varphi_2(K)=varphi_2(K^a) tag{4}$$
    where $K^a$ is the image of $K$ under the $a$-th power homomorphism $K to K:k mapsto k^a$. Since $varphi_2$ is injective, this is only possible if the $a$-th power homomorphism is surjective, which happens iff $a=pm 1$. We can thus see that
    $$chi((h,k))=(sigma^{-1}(h),k^a)$$
    is a 2-sided inverse of $psi$.



  • Assume $K=langle x rangle cong Z_n$ is finite cyclic of order $n$, and denote the orders of the cyclic groups $varphi_1(K),varphi_2(K)$ by $m$. I believe that $(a,n) = 1$ so that there is some integer $b$ such that $ab equiv 1 pmod{n}$. If we have this, we can see that
    $$chi((h,k))=(sigma^{-1}(h),k^b) $$
    is a 2-sided inverse of $psi$. However, all I could show is the following.




    1. Obviously, $m|n$.

    2. Since $varphi_1(K)=langle varphi_1(x) rangle$ and $varphi_2(K)=langle varphi_2(K) rangle$ we have $|varphi_1(x)|=|varphi_2(x)|$. According to equation (3) $varphi_2(K)$ is also generated by $varphi_2(x^a)$, so that $|varphi_2(x^a)|=|varphi_2(x)|$ which gives $(a,m)=1$.

    3. Raising equation $(2)$ to the power of $|x^a|=frac{n}{(a,n)}$ gives $1=varphi_2(1)=varphi_2((x^{a})^{frac{n}{(a,n)}})=varphi_2(x^a)^{frac{n}{(a,n)}}$, so that $m| frac{n}{(a,n)}$.


    4. From the first occurrence of "$a$" in equation $(2)$, and the fact that $varphi_2(x^a)=varphi_2(x)^a$ we can see that $a$ may shifted by any multiple of $m$.







My questions:




  1. Are there any flaws with my proof?


  2. It seems that in the infinite case, it is only necessary to assume one of the $varphi_i$'s to be injective. Is this true?


  3. Is it possible to prove that "$a$" can be chosen coprime with $n$?



Thank you!










share|cite|improve this question









$endgroup$

















    5












    $begingroup$


    I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 184 Exercise 6):




    Assume that $K$ is a cyclic group, $H$ is an arbitrary group and $varphi_1$ and $varphi_2$ are homomorphisms from $K$ into $text{Aut}(H)$ such that $varphi_1(K)$ and $varphi_2(K)$ are subgroups of $text{Aut}(H)$. If $K$ is infinite assume $varphi_1$ and $varphi_2$ are injective. Prove by constructing an explicit isomorphism that $H rtimes_{varphi_1} K cong H rtimes_{varphi_2} K$ (in particular, if the subgroups $varphi_1(K)$ and $varphi_2(K)$ are equal in $text{Aut}(H)$, then the resulting semidirect products are isomorphic). [Suppose $sigma varphi_1(K) sigma^{-1}=varphi_2(K)$ so that for some $a in mathbb{Z}$ we have $sigma varphi_1(k) sigma^{-1}=varphi_2(k)^a$ for all $k in K$. Show that the map $psi:H rtimes_{varphi_1} K to H rtimes_{varphi_2} K$ defined by $psi((h,k))=(sigma(h),k^a)$ is a homomorphism. Show $psi$ is bijective by constructing a 2-sided inverse.]






    My attempt:
    Let $K=langle x rangle$, and suppose $sigma in text{Aut}(H)$ conjugates $varphi_1(K)$ to $varphi_2(K)$, that is



    $$sigma varphi_1(K) sigma^{-1}=varphi_2(K) tag{1}.$$



    From this we find
    $$sigma varphi_1(x) sigma^{-1}=varphi_2(x^a) tag{2} $$
    for some $a in mathbb{Z}$. Since all elements of $K$ are powers $x^n$ of $x$, raising this equality to the $n$-th power gives
    $$forall k in K:sigma varphi_1(k) sigma^{-1}=varphi_2(k^a)=varphi_2(k)^a tag{3}.$$



    We now prove that the suggested $psi$ is a group homomorphism:



    begin{equation}
    begin{split}
    psi((h_1,k_1)(h_2,k_2))&=psi(h_1 varphi_1(k_1)(h_2),k_1k_2)=(sigma(h_1 varphi_1(k_1)(h_2)),(k_1k_2)^a)\
    &=(sigma(h_1)(sigma varphi_1(k_1))(h_2),k_1^a k_2^a)=(sigma(h_1)(varphi_2(k_1^a) sigma)(h_2),k_1^ak_2^a)\
    &=(sigma(h_1),k_1^a)(sigma(h_2),k_2^a)=psi((h_1,k_1))psi((h_2,k_2))
    end{split}
    end{equation}



    Where we have used the fact that $K$ is abelian, alongside with the homomorphism law for $sigma$ and equation $(3)$. We're left with showing that $psi$ has a 2-sided inverse, which will be done in two cases:






    • Assume $K=langle x rangle$ is infinite cyclic. Property $(3)$ gives
      $$varphi_2(K)=varphi_2(K^a) tag{4}$$
      where $K^a$ is the image of $K$ under the $a$-th power homomorphism $K to K:k mapsto k^a$. Since $varphi_2$ is injective, this is only possible if the $a$-th power homomorphism is surjective, which happens iff $a=pm 1$. We can thus see that
      $$chi((h,k))=(sigma^{-1}(h),k^a)$$
      is a 2-sided inverse of $psi$.



    • Assume $K=langle x rangle cong Z_n$ is finite cyclic of order $n$, and denote the orders of the cyclic groups $varphi_1(K),varphi_2(K)$ by $m$. I believe that $(a,n) = 1$ so that there is some integer $b$ such that $ab equiv 1 pmod{n}$. If we have this, we can see that
      $$chi((h,k))=(sigma^{-1}(h),k^b) $$
      is a 2-sided inverse of $psi$. However, all I could show is the following.




      1. Obviously, $m|n$.

      2. Since $varphi_1(K)=langle varphi_1(x) rangle$ and $varphi_2(K)=langle varphi_2(K) rangle$ we have $|varphi_1(x)|=|varphi_2(x)|$. According to equation (3) $varphi_2(K)$ is also generated by $varphi_2(x^a)$, so that $|varphi_2(x^a)|=|varphi_2(x)|$ which gives $(a,m)=1$.

      3. Raising equation $(2)$ to the power of $|x^a|=frac{n}{(a,n)}$ gives $1=varphi_2(1)=varphi_2((x^{a})^{frac{n}{(a,n)}})=varphi_2(x^a)^{frac{n}{(a,n)}}$, so that $m| frac{n}{(a,n)}$.


      4. From the first occurrence of "$a$" in equation $(2)$, and the fact that $varphi_2(x^a)=varphi_2(x)^a$ we can see that $a$ may shifted by any multiple of $m$.







    My questions:




    1. Are there any flaws with my proof?


    2. It seems that in the infinite case, it is only necessary to assume one of the $varphi_i$'s to be injective. Is this true?


    3. Is it possible to prove that "$a$" can be chosen coprime with $n$?



    Thank you!










    share|cite|improve this question









    $endgroup$















      5












      5








      5





      $begingroup$


      I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 184 Exercise 6):




      Assume that $K$ is a cyclic group, $H$ is an arbitrary group and $varphi_1$ and $varphi_2$ are homomorphisms from $K$ into $text{Aut}(H)$ such that $varphi_1(K)$ and $varphi_2(K)$ are subgroups of $text{Aut}(H)$. If $K$ is infinite assume $varphi_1$ and $varphi_2$ are injective. Prove by constructing an explicit isomorphism that $H rtimes_{varphi_1} K cong H rtimes_{varphi_2} K$ (in particular, if the subgroups $varphi_1(K)$ and $varphi_2(K)$ are equal in $text{Aut}(H)$, then the resulting semidirect products are isomorphic). [Suppose $sigma varphi_1(K) sigma^{-1}=varphi_2(K)$ so that for some $a in mathbb{Z}$ we have $sigma varphi_1(k) sigma^{-1}=varphi_2(k)^a$ for all $k in K$. Show that the map $psi:H rtimes_{varphi_1} K to H rtimes_{varphi_2} K$ defined by $psi((h,k))=(sigma(h),k^a)$ is a homomorphism. Show $psi$ is bijective by constructing a 2-sided inverse.]






      My attempt:
      Let $K=langle x rangle$, and suppose $sigma in text{Aut}(H)$ conjugates $varphi_1(K)$ to $varphi_2(K)$, that is



      $$sigma varphi_1(K) sigma^{-1}=varphi_2(K) tag{1}.$$



      From this we find
      $$sigma varphi_1(x) sigma^{-1}=varphi_2(x^a) tag{2} $$
      for some $a in mathbb{Z}$. Since all elements of $K$ are powers $x^n$ of $x$, raising this equality to the $n$-th power gives
      $$forall k in K:sigma varphi_1(k) sigma^{-1}=varphi_2(k^a)=varphi_2(k)^a tag{3}.$$



      We now prove that the suggested $psi$ is a group homomorphism:



      begin{equation}
      begin{split}
      psi((h_1,k_1)(h_2,k_2))&=psi(h_1 varphi_1(k_1)(h_2),k_1k_2)=(sigma(h_1 varphi_1(k_1)(h_2)),(k_1k_2)^a)\
      &=(sigma(h_1)(sigma varphi_1(k_1))(h_2),k_1^a k_2^a)=(sigma(h_1)(varphi_2(k_1^a) sigma)(h_2),k_1^ak_2^a)\
      &=(sigma(h_1),k_1^a)(sigma(h_2),k_2^a)=psi((h_1,k_1))psi((h_2,k_2))
      end{split}
      end{equation}



      Where we have used the fact that $K$ is abelian, alongside with the homomorphism law for $sigma$ and equation $(3)$. We're left with showing that $psi$ has a 2-sided inverse, which will be done in two cases:






      • Assume $K=langle x rangle$ is infinite cyclic. Property $(3)$ gives
        $$varphi_2(K)=varphi_2(K^a) tag{4}$$
        where $K^a$ is the image of $K$ under the $a$-th power homomorphism $K to K:k mapsto k^a$. Since $varphi_2$ is injective, this is only possible if the $a$-th power homomorphism is surjective, which happens iff $a=pm 1$. We can thus see that
        $$chi((h,k))=(sigma^{-1}(h),k^a)$$
        is a 2-sided inverse of $psi$.



      • Assume $K=langle x rangle cong Z_n$ is finite cyclic of order $n$, and denote the orders of the cyclic groups $varphi_1(K),varphi_2(K)$ by $m$. I believe that $(a,n) = 1$ so that there is some integer $b$ such that $ab equiv 1 pmod{n}$. If we have this, we can see that
        $$chi((h,k))=(sigma^{-1}(h),k^b) $$
        is a 2-sided inverse of $psi$. However, all I could show is the following.




        1. Obviously, $m|n$.

        2. Since $varphi_1(K)=langle varphi_1(x) rangle$ and $varphi_2(K)=langle varphi_2(K) rangle$ we have $|varphi_1(x)|=|varphi_2(x)|$. According to equation (3) $varphi_2(K)$ is also generated by $varphi_2(x^a)$, so that $|varphi_2(x^a)|=|varphi_2(x)|$ which gives $(a,m)=1$.

        3. Raising equation $(2)$ to the power of $|x^a|=frac{n}{(a,n)}$ gives $1=varphi_2(1)=varphi_2((x^{a})^{frac{n}{(a,n)}})=varphi_2(x^a)^{frac{n}{(a,n)}}$, so that $m| frac{n}{(a,n)}$.


        4. From the first occurrence of "$a$" in equation $(2)$, and the fact that $varphi_2(x^a)=varphi_2(x)^a$ we can see that $a$ may shifted by any multiple of $m$.







      My questions:




      1. Are there any flaws with my proof?


      2. It seems that in the infinite case, it is only necessary to assume one of the $varphi_i$'s to be injective. Is this true?


      3. Is it possible to prove that "$a$" can be chosen coprime with $n$?



      Thank you!










      share|cite|improve this question









      $endgroup$




      I want to solve the following problem from Dummit & Foote's Abstract Algebra text (p. 184 Exercise 6):




      Assume that $K$ is a cyclic group, $H$ is an arbitrary group and $varphi_1$ and $varphi_2$ are homomorphisms from $K$ into $text{Aut}(H)$ such that $varphi_1(K)$ and $varphi_2(K)$ are subgroups of $text{Aut}(H)$. If $K$ is infinite assume $varphi_1$ and $varphi_2$ are injective. Prove by constructing an explicit isomorphism that $H rtimes_{varphi_1} K cong H rtimes_{varphi_2} K$ (in particular, if the subgroups $varphi_1(K)$ and $varphi_2(K)$ are equal in $text{Aut}(H)$, then the resulting semidirect products are isomorphic). [Suppose $sigma varphi_1(K) sigma^{-1}=varphi_2(K)$ so that for some $a in mathbb{Z}$ we have $sigma varphi_1(k) sigma^{-1}=varphi_2(k)^a$ for all $k in K$. Show that the map $psi:H rtimes_{varphi_1} K to H rtimes_{varphi_2} K$ defined by $psi((h,k))=(sigma(h),k^a)$ is a homomorphism. Show $psi$ is bijective by constructing a 2-sided inverse.]






      My attempt:
      Let $K=langle x rangle$, and suppose $sigma in text{Aut}(H)$ conjugates $varphi_1(K)$ to $varphi_2(K)$, that is



      $$sigma varphi_1(K) sigma^{-1}=varphi_2(K) tag{1}.$$



      From this we find
      $$sigma varphi_1(x) sigma^{-1}=varphi_2(x^a) tag{2} $$
      for some $a in mathbb{Z}$. Since all elements of $K$ are powers $x^n$ of $x$, raising this equality to the $n$-th power gives
      $$forall k in K:sigma varphi_1(k) sigma^{-1}=varphi_2(k^a)=varphi_2(k)^a tag{3}.$$



      We now prove that the suggested $psi$ is a group homomorphism:



      begin{equation}
      begin{split}
      psi((h_1,k_1)(h_2,k_2))&=psi(h_1 varphi_1(k_1)(h_2),k_1k_2)=(sigma(h_1 varphi_1(k_1)(h_2)),(k_1k_2)^a)\
      &=(sigma(h_1)(sigma varphi_1(k_1))(h_2),k_1^a k_2^a)=(sigma(h_1)(varphi_2(k_1^a) sigma)(h_2),k_1^ak_2^a)\
      &=(sigma(h_1),k_1^a)(sigma(h_2),k_2^a)=psi((h_1,k_1))psi((h_2,k_2))
      end{split}
      end{equation}



      Where we have used the fact that $K$ is abelian, alongside with the homomorphism law for $sigma$ and equation $(3)$. We're left with showing that $psi$ has a 2-sided inverse, which will be done in two cases:






      • Assume $K=langle x rangle$ is infinite cyclic. Property $(3)$ gives
        $$varphi_2(K)=varphi_2(K^a) tag{4}$$
        where $K^a$ is the image of $K$ under the $a$-th power homomorphism $K to K:k mapsto k^a$. Since $varphi_2$ is injective, this is only possible if the $a$-th power homomorphism is surjective, which happens iff $a=pm 1$. We can thus see that
        $$chi((h,k))=(sigma^{-1}(h),k^a)$$
        is a 2-sided inverse of $psi$.



      • Assume $K=langle x rangle cong Z_n$ is finite cyclic of order $n$, and denote the orders of the cyclic groups $varphi_1(K),varphi_2(K)$ by $m$. I believe that $(a,n) = 1$ so that there is some integer $b$ such that $ab equiv 1 pmod{n}$. If we have this, we can see that
        $$chi((h,k))=(sigma^{-1}(h),k^b) $$
        is a 2-sided inverse of $psi$. However, all I could show is the following.




        1. Obviously, $m|n$.

        2. Since $varphi_1(K)=langle varphi_1(x) rangle$ and $varphi_2(K)=langle varphi_2(K) rangle$ we have $|varphi_1(x)|=|varphi_2(x)|$. According to equation (3) $varphi_2(K)$ is also generated by $varphi_2(x^a)$, so that $|varphi_2(x^a)|=|varphi_2(x)|$ which gives $(a,m)=1$.

        3. Raising equation $(2)$ to the power of $|x^a|=frac{n}{(a,n)}$ gives $1=varphi_2(1)=varphi_2((x^{a})^{frac{n}{(a,n)}})=varphi_2(x^a)^{frac{n}{(a,n)}}$, so that $m| frac{n}{(a,n)}$.


        4. From the first occurrence of "$a$" in equation $(2)$, and the fact that $varphi_2(x^a)=varphi_2(x)^a$ we can see that $a$ may shifted by any multiple of $m$.







      My questions:




      1. Are there any flaws with my proof?


      2. It seems that in the infinite case, it is only necessary to assume one of the $varphi_i$'s to be injective. Is this true?


      3. Is it possible to prove that "$a$" can be chosen coprime with $n$?



      Thank you!







      abstract-algebra group-theory semidirect-product






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      asked Jan 6 '15 at 15:11









      user1337user1337

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          $begingroup$

          If $m$ is a divisor of $n$ then the natural map $mathbb{Z}_nrightarrowmathbb{Z}_m$ induces a surjective homomorphism $U(n)rightarrow U(m)$, see here for a proof. You have already noticed that $(a,m)=1$. By the aforementioned fact, there exists $a'$ such that $a'equiv a;(mathrm{mod};m)$ and $(a',n)=1$. Notice $varphi_2(k)^a=varphi_2(k)^{a'+lm}=varphi_2(k)^{a'}$. Therefore by replacing $a$ with $a'$ you can assume that $(a,n)=1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I can't understand why there exists $a'$ such that $(a',n)=1$. Could you explain me more precisely?
            $endgroup$
            – Pearl
            Dec 2 '18 at 22:56












          • $begingroup$
            @Pearl: $(a,m)=1$ therefore $ain U(m)$. Since the map $U(n)rightarrow U(m)$ is surjective we can take a preimage of $a$, call it $a'$. Since $a'in U(n)$ then $(a',n)=1$. By the way the map $U(n)rightarrow U(m)$ is defined we also have $aequiv a';(m)$.
            $endgroup$
            – Mec
            Dec 5 '18 at 23:17











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          1 Answer
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          active

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          $begingroup$

          If $m$ is a divisor of $n$ then the natural map $mathbb{Z}_nrightarrowmathbb{Z}_m$ induces a surjective homomorphism $U(n)rightarrow U(m)$, see here for a proof. You have already noticed that $(a,m)=1$. By the aforementioned fact, there exists $a'$ such that $a'equiv a;(mathrm{mod};m)$ and $(a',n)=1$. Notice $varphi_2(k)^a=varphi_2(k)^{a'+lm}=varphi_2(k)^{a'}$. Therefore by replacing $a$ with $a'$ you can assume that $(a,n)=1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I can't understand why there exists $a'$ such that $(a',n)=1$. Could you explain me more precisely?
            $endgroup$
            – Pearl
            Dec 2 '18 at 22:56












          • $begingroup$
            @Pearl: $(a,m)=1$ therefore $ain U(m)$. Since the map $U(n)rightarrow U(m)$ is surjective we can take a preimage of $a$, call it $a'$. Since $a'in U(n)$ then $(a',n)=1$. By the way the map $U(n)rightarrow U(m)$ is defined we also have $aequiv a';(m)$.
            $endgroup$
            – Mec
            Dec 5 '18 at 23:17
















          0












          $begingroup$

          If $m$ is a divisor of $n$ then the natural map $mathbb{Z}_nrightarrowmathbb{Z}_m$ induces a surjective homomorphism $U(n)rightarrow U(m)$, see here for a proof. You have already noticed that $(a,m)=1$. By the aforementioned fact, there exists $a'$ such that $a'equiv a;(mathrm{mod};m)$ and $(a',n)=1$. Notice $varphi_2(k)^a=varphi_2(k)^{a'+lm}=varphi_2(k)^{a'}$. Therefore by replacing $a$ with $a'$ you can assume that $(a,n)=1$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            I can't understand why there exists $a'$ such that $(a',n)=1$. Could you explain me more precisely?
            $endgroup$
            – Pearl
            Dec 2 '18 at 22:56












          • $begingroup$
            @Pearl: $(a,m)=1$ therefore $ain U(m)$. Since the map $U(n)rightarrow U(m)$ is surjective we can take a preimage of $a$, call it $a'$. Since $a'in U(n)$ then $(a',n)=1$. By the way the map $U(n)rightarrow U(m)$ is defined we also have $aequiv a';(m)$.
            $endgroup$
            – Mec
            Dec 5 '18 at 23:17














          0












          0








          0





          $begingroup$

          If $m$ is a divisor of $n$ then the natural map $mathbb{Z}_nrightarrowmathbb{Z}_m$ induces a surjective homomorphism $U(n)rightarrow U(m)$, see here for a proof. You have already noticed that $(a,m)=1$. By the aforementioned fact, there exists $a'$ such that $a'equiv a;(mathrm{mod};m)$ and $(a',n)=1$. Notice $varphi_2(k)^a=varphi_2(k)^{a'+lm}=varphi_2(k)^{a'}$. Therefore by replacing $a$ with $a'$ you can assume that $(a,n)=1$.






          share|cite|improve this answer









          $endgroup$



          If $m$ is a divisor of $n$ then the natural map $mathbb{Z}_nrightarrowmathbb{Z}_m$ induces a surjective homomorphism $U(n)rightarrow U(m)$, see here for a proof. You have already noticed that $(a,m)=1$. By the aforementioned fact, there exists $a'$ such that $a'equiv a;(mathrm{mod};m)$ and $(a',n)=1$. Notice $varphi_2(k)^a=varphi_2(k)^{a'+lm}=varphi_2(k)^{a'}$. Therefore by replacing $a$ with $a'$ you can assume that $(a,n)=1$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 29 '18 at 22:38









          MecMec

          2,27021529




          2,27021529












          • $begingroup$
            I can't understand why there exists $a'$ such that $(a',n)=1$. Could you explain me more precisely?
            $endgroup$
            – Pearl
            Dec 2 '18 at 22:56












          • $begingroup$
            @Pearl: $(a,m)=1$ therefore $ain U(m)$. Since the map $U(n)rightarrow U(m)$ is surjective we can take a preimage of $a$, call it $a'$. Since $a'in U(n)$ then $(a',n)=1$. By the way the map $U(n)rightarrow U(m)$ is defined we also have $aequiv a';(m)$.
            $endgroup$
            – Mec
            Dec 5 '18 at 23:17


















          • $begingroup$
            I can't understand why there exists $a'$ such that $(a',n)=1$. Could you explain me more precisely?
            $endgroup$
            – Pearl
            Dec 2 '18 at 22:56












          • $begingroup$
            @Pearl: $(a,m)=1$ therefore $ain U(m)$. Since the map $U(n)rightarrow U(m)$ is surjective we can take a preimage of $a$, call it $a'$. Since $a'in U(n)$ then $(a',n)=1$. By the way the map $U(n)rightarrow U(m)$ is defined we also have $aequiv a';(m)$.
            $endgroup$
            – Mec
            Dec 5 '18 at 23:17
















          $begingroup$
          I can't understand why there exists $a'$ such that $(a',n)=1$. Could you explain me more precisely?
          $endgroup$
          – Pearl
          Dec 2 '18 at 22:56






          $begingroup$
          I can't understand why there exists $a'$ such that $(a',n)=1$. Could you explain me more precisely?
          $endgroup$
          – Pearl
          Dec 2 '18 at 22:56














          $begingroup$
          @Pearl: $(a,m)=1$ therefore $ain U(m)$. Since the map $U(n)rightarrow U(m)$ is surjective we can take a preimage of $a$, call it $a'$. Since $a'in U(n)$ then $(a',n)=1$. By the way the map $U(n)rightarrow U(m)$ is defined we also have $aequiv a';(m)$.
          $endgroup$
          – Mec
          Dec 5 '18 at 23:17




          $begingroup$
          @Pearl: $(a,m)=1$ therefore $ain U(m)$. Since the map $U(n)rightarrow U(m)$ is surjective we can take a preimage of $a$, call it $a'$. Since $a'in U(n)$ then $(a',n)=1$. By the way the map $U(n)rightarrow U(m)$ is defined we also have $aequiv a';(m)$.
          $endgroup$
          – Mec
          Dec 5 '18 at 23:17


















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