which of the following is a subgroup of $mathbb{Z}^2$











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consider the group $mathbb{Z}^2 ={(a,b)| a,b in mathbb{Z} }$ under component wise addition.



then which of the following is a subgroup of $mathbb{Z}^2$



$a)$ ${(a,b) in mathbb{Z}^2 | ab=0}$



$b)$ ${(a,b) in mathbb{Z}^2 | 3a +2b=15}$



$c)$ ${(a,b) in mathbb{Z}^2 |text{7 divides ab}}$



$d)$ ${(a,b) in mathbb{Z}^2 |text{2 divides a and 3 divides b}}$



My attempts :



I know that under component wise addition will be $(a,b) + (a,b)= (2a,2b)$



Here $2$ is Divide $a$. Now i can not proceed further



Any hints/solution will be appreciated



thanks u






abstract-algebra







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    Do you know how to check/prove that a subset of a group is a subgroup?
    – Dionel Jaime
    22 hours ago















up vote
-1
down vote

favorite












consider the group $mathbb{Z}^2 ={(a,b)| a,b in mathbb{Z} }$ under component wise addition.



then which of the following is a subgroup of $mathbb{Z}^2$



$a)$ ${(a,b) in mathbb{Z}^2 | ab=0}$



$b)$ ${(a,b) in mathbb{Z}^2 | 3a +2b=15}$



$c)$ ${(a,b) in mathbb{Z}^2 |text{7 divides ab}}$



$d)$ ${(a,b) in mathbb{Z}^2 |text{2 divides a and 3 divides b}}$



My attempts :



I know that under component wise addition will be $(a,b) + (a,b)= (2a,2b)$



Here $2$ is Divide $a$. Now i can not proceed further



Any hints/solution will be appreciated



thanks u






abstract-algebra







share|cite|improve this question


















  • 1




    Do you know how to check/prove that a subset of a group is a subgroup?
    – Dionel Jaime
    22 hours ago













up vote
-1
down vote

favorite









up vote
-1
down vote

favorite











consider the group $mathbb{Z}^2 ={(a,b)| a,b in mathbb{Z} }$ under component wise addition.



then which of the following is a subgroup of $mathbb{Z}^2$



$a)$ ${(a,b) in mathbb{Z}^2 | ab=0}$



$b)$ ${(a,b) in mathbb{Z}^2 | 3a +2b=15}$



$c)$ ${(a,b) in mathbb{Z}^2 |text{7 divides ab}}$



$d)$ ${(a,b) in mathbb{Z}^2 |text{2 divides a and 3 divides b}}$



My attempts :



I know that under component wise addition will be $(a,b) + (a,b)= (2a,2b)$



Here $2$ is Divide $a$. Now i can not proceed further



Any hints/solution will be appreciated



thanks u






abstract-algebra







share|cite|improve this question













consider the group $mathbb{Z}^2 ={(a,b)| a,b in mathbb{Z} }$ under component wise addition.



then which of the following is a subgroup of $mathbb{Z}^2$



$a)$ ${(a,b) in mathbb{Z}^2 | ab=0}$



$b)$ ${(a,b) in mathbb{Z}^2 | 3a +2b=15}$



$c)$ ${(a,b) in mathbb{Z}^2 |text{7 divides ab}}$



$d)$ ${(a,b) in mathbb{Z}^2 |text{2 divides a and 3 divides b}}$



My attempts :



I know that under component wise addition will be $(a,b) + (a,b)= (2a,2b)$



Here $2$ is Divide $a$. Now i can not proceed further



Any hints/solution will be appreciated



thanks u






abstract-algebra




abstract-algebra






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asked 22 hours ago









jasmine

1,278316




1,278316








  • 1




    Do you know how to check/prove that a subset of a group is a subgroup?
    – Dionel Jaime
    22 hours ago














  • 1




    Do you know how to check/prove that a subset of a group is a subgroup?
    – Dionel Jaime
    22 hours ago








1




1




Do you know how to check/prove that a subset of a group is a subgroup?
– Dionel Jaime
22 hours ago




Do you know how to check/prove that a subset of a group is a subgroup?
– Dionel Jaime
22 hours ago










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Hint : A non empty subset $H$ of $G$ is a subgroup of $G $ iff for every $x,y in H$, $xy^{-1} in H $.






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    Hint : A non empty subset $H$ of $G$ is a subgroup of $G $ iff for every $x,y in H$, $xy^{-1} in H $.






    share|cite|improve this answer

























      up vote
      1
      down vote













      Hint : A non empty subset $H$ of $G$ is a subgroup of $G $ iff for every $x,y in H$, $xy^{-1} in H $.






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        Hint : A non empty subset $H$ of $G$ is a subgroup of $G $ iff for every $x,y in H$, $xy^{-1} in H $.






        share|cite|improve this answer












        Hint : A non empty subset $H$ of $G$ is a subgroup of $G $ iff for every $x,y in H$, $xy^{-1} in H $.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 19 hours ago









        Thomas Shelby

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