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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-1
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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
1
Do you know how to check/prove that a subset of a group is a subgroup?
– Dionel Jaime
22 hours ago
add a comment |
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
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
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
add a comment |
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
add a comment |
1 Answer
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1
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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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1 Answer
1
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oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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 $.
add a comment |
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 $.
add a comment |
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 $.
Hint : A non empty subset $H$ of $G$ is a subgroup of $G $ iff for every $x,y in H$, $xy^{-1} in H $.
answered 19 hours ago
Thomas Shelby
356
356
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1
Do you know how to check/prove that a subset of a group is a subgroup?
– Dionel Jaime
22 hours ago