Prove that $BbbZ×BbbZ$ and $BbbZ×BbbZ×BbbZ$ is not isomorphic groups. [closed]

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
-3
down vote

favorite












Considering them as rings, I can prove that they are not isomorphic as they have different nos of idempotent elements , but in case of group isomorphism I am unable to prove it.




abstract-algebra group-isomorphism




share|cite|improve this question













closed as off-topic by amWhy, anomaly, Dietrich Burde, Xander Henderson, Alexander Gruber♦ Jul 29 at 5:50


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, anomaly, Dietrich Burde, Xander Henderson, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.












  • Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Jul 28 at 13:26














up vote
-3
down vote

favorite












Considering them as rings, I can prove that they are not isomorphic as they have different nos of idempotent elements , but in case of group isomorphism I am unable to prove it.




abstract-algebra group-isomorphism




share|cite|improve this question













closed as off-topic by amWhy, anomaly, Dietrich Burde, Xander Henderson, Alexander Gruber♦ Jul 29 at 5:50


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, anomaly, Dietrich Burde, Xander Henderson, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.












  • Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Jul 28 at 13:26












up vote
-3
down vote

favorite









up vote
-3
down vote

favorite











Considering them as rings, I can prove that they are not isomorphic as they have different nos of idempotent elements , but in case of group isomorphism I am unable to prove it.




abstract-algebra group-isomorphism




share|cite|improve this question













Considering them as rings, I can prove that they are not isomorphic as they have different nos of idempotent elements , but in case of group isomorphism I am unable to prove it.





abstract-algebra group-isomorphism





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 28 at 13:32









Sou

2,6792820




2,6792820









asked Jul 28 at 13:20









A. B.

1




1




closed as off-topic by amWhy, anomaly, Dietrich Burde, Xander Henderson, Alexander Gruber♦ Jul 29 at 5:50


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, anomaly, Dietrich Burde, Xander Henderson, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.




closed as off-topic by amWhy, anomaly, Dietrich Burde, Xander Henderson, Alexander Gruber♦ Jul 29 at 5:50


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – amWhy, anomaly, Dietrich Burde, Xander Henderson, Alexander Gruber
If this question can be reworded to fit the rules in the help center, please edit the question.











  • Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Jul 28 at 13:26
















  • Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
    – José Carlos Santos
    Jul 28 at 13:26















Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 28 at 13:26




Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– José Carlos Santos
Jul 28 at 13:26










3 Answers
3






active

oldest

votes

















up vote
3
down vote













Hint: consider the possible cardinalities of sets of odd elements such that the sum of any two is odd. Alternatively, show that if $G_1$, $G_2$ are isomorphic abelian groups, then $G_1/2G_1$ and $G_2/2G_2$ are isomorphic.






share|cite|improve this answer




























    up vote
    2
    down vote













    Hint:



    $mathbb Ztimesmathbb Z$ can be generated by $2$ elements.



    $mathbb Ztimesmathbb Ztimesmathbb Z$ cannot be generated by $2$ elements.



    (Prove that for two elements $(a,b,c)$ and $(u,v,w)$ the set $n(a,b,c)+m(u,v,w)mid n,minmathbb Z$ is Always a proper subset of $mathbb Ztimesmathbb Ztimesmathbb Z$)






    share|cite|improve this answer



















    • 1




      Of course the second line requires proof.
      – GEdgar
      Jul 28 at 13:37

















    up vote
    1
    down vote













    That results from the Invariant Basis Number property in commutative rings: they're frr $bf Z$-modules, with a basis of $2$ and $3$ vectors respectively.






    share|cite|improve this answer




























      3 Answers
      3






      active

      oldest

      votes








      3 Answers
      3






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      3
      down vote













      Hint: consider the possible cardinalities of sets of odd elements such that the sum of any two is odd. Alternatively, show that if $G_1$, $G_2$ are isomorphic abelian groups, then $G_1/2G_1$ and $G_2/2G_2$ are isomorphic.






      share|cite|improve this answer

























        up vote
        3
        down vote













        Hint: consider the possible cardinalities of sets of odd elements such that the sum of any two is odd. Alternatively, show that if $G_1$, $G_2$ are isomorphic abelian groups, then $G_1/2G_1$ and $G_2/2G_2$ are isomorphic.






        share|cite|improve this answer























          up vote
          3
          down vote










          up vote
          3
          down vote









          Hint: consider the possible cardinalities of sets of odd elements such that the sum of any two is odd. Alternatively, show that if $G_1$, $G_2$ are isomorphic abelian groups, then $G_1/2G_1$ and $G_2/2G_2$ are isomorphic.






          share|cite|improve this answer













          Hint: consider the possible cardinalities of sets of odd elements such that the sum of any two is odd. Alternatively, show that if $G_1$, $G_2$ are isomorphic abelian groups, then $G_1/2G_1$ and $G_2/2G_2$ are isomorphic.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Jul 28 at 13:36









          tomasz

          22.7k22975




          22.7k22975




















              up vote
              2
              down vote













              Hint:



              $mathbb Ztimesmathbb Z$ can be generated by $2$ elements.



              $mathbb Ztimesmathbb Ztimesmathbb Z$ cannot be generated by $2$ elements.



              (Prove that for two elements $(a,b,c)$ and $(u,v,w)$ the set $n(a,b,c)+m(u,v,w)mid n,minmathbb Z$ is Always a proper subset of $mathbb Ztimesmathbb Ztimesmathbb Z$)






              share|cite|improve this answer



















              • 1




                Of course the second line requires proof.
                – GEdgar
                Jul 28 at 13:37














              up vote
              2
              down vote













              Hint:



              $mathbb Ztimesmathbb Z$ can be generated by $2$ elements.



              $mathbb Ztimesmathbb Ztimesmathbb Z$ cannot be generated by $2$ elements.



              (Prove that for two elements $(a,b,c)$ and $(u,v,w)$ the set $n(a,b,c)+m(u,v,w)mid n,minmathbb Z$ is Always a proper subset of $mathbb Ztimesmathbb Ztimesmathbb Z$)






              share|cite|improve this answer



















              • 1




                Of course the second line requires proof.
                – GEdgar
                Jul 28 at 13:37












              up vote
              2
              down vote










              up vote
              2
              down vote









              Hint:



              $mathbb Ztimesmathbb Z$ can be generated by $2$ elements.



              $mathbb Ztimesmathbb Ztimesmathbb Z$ cannot be generated by $2$ elements.



              (Prove that for two elements $(a,b,c)$ and $(u,v,w)$ the set $n(a,b,c)+m(u,v,w)mid n,minmathbb Z$ is Always a proper subset of $mathbb Ztimesmathbb Ztimesmathbb Z$)






              share|cite|improve this answer















              Hint:



              $mathbb Ztimesmathbb Z$ can be generated by $2$ elements.



              $mathbb Ztimesmathbb Ztimesmathbb Z$ cannot be generated by $2$ elements.



              (Prove that for two elements $(a,b,c)$ and $(u,v,w)$ the set $n(a,b,c)+m(u,v,w)mid n,minmathbb Z$ is Always a proper subset of $mathbb Ztimesmathbb Ztimesmathbb Z$)







              share|cite|improve this answer















              share|cite|improve this answer



              share|cite|improve this answer








              edited Jul 28 at 13:58


























              answered Jul 28 at 13:26









              Vera

              1,696313




              1,696313







              • 1




                Of course the second line requires proof.
                – GEdgar
                Jul 28 at 13:37












              • 1




                Of course the second line requires proof.
                – GEdgar
                Jul 28 at 13:37







              1




              1




              Of course the second line requires proof.
              – GEdgar
              Jul 28 at 13:37




              Of course the second line requires proof.
              – GEdgar
              Jul 28 at 13:37










              up vote
              1
              down vote













              That results from the Invariant Basis Number property in commutative rings: they're frr $bf Z$-modules, with a basis of $2$ and $3$ vectors respectively.






              share|cite|improve this answer

























                up vote
                1
                down vote













                That results from the Invariant Basis Number property in commutative rings: they're frr $bf Z$-modules, with a basis of $2$ and $3$ vectors respectively.






                share|cite|improve this answer























                  up vote
                  1
                  down vote










                  up vote
                  1
                  down vote









                  That results from the Invariant Basis Number property in commutative rings: they're frr $bf Z$-modules, with a basis of $2$ and $3$ vectors respectively.






                  share|cite|improve this answer













                  That results from the Invariant Basis Number property in commutative rings: they're frr $bf Z$-modules, with a basis of $2$ and $3$ vectors respectively.







                  share|cite|improve this answer













                  share|cite|improve this answer



                  share|cite|improve this answer











                  answered Jul 28 at 13:27









                  Bernard

                  110k635102




                  110k635102












                      Comments

                      Post a Comment

                      Popular posts from this blog

                      What is the equation of a 3D cone with generalised tilt?

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
                      Read more

                      Relationship between determinant of matrix and determinant of adjoint?

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
                      Read more

                      Color the edges and diagonals of a regular polygon

                      Image
                      Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
                      Read more