Mixing
On isomorphism of quotient groups of free abelian groups of finite rank
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Consider the free abelian group $mathbb Z^n$, with elements considered as row vectors. For every $Ain M_rtimes n(mathbb Z)$ , let $K_A$ be the subgroup of $mathbb Z^n$ generated by the row vectors of $A$ . Now let $Ain M_rtimes n(mathbb Z)$ and $B:=PAQ$ where $Pin GL_rtimes r(mathbb Z), Q in GL_ntimes n(mathbb Z)$. Then how to prove that $mathbb Z^n/K_A $ and $mathbb Z^n/K_B$ are isomorphic as groups ?
I think the map $f: mathbb Z^n/K_A to mathbb Z^n/K_B$ defined as $f(bar a +K_A)=bar a Q+K_B, forall bar a in mathbb Z^n$ is an isomorphism, but I am not sure (I can't even show whether this map is well-defined or not).
Please help.
modules abelian-groups group-isomorphism free-modules free-abelian-group
asked Jul 28 at 17:11
user521337
606
add a comment |Â
up vote
0
down vote
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Consider the free abelian group $mathbb Z^n$, with elements considered as row vectors. For every $Ain M_rtimes n(mathbb Z)$ , let $K_A$ be the subgroup of $mathbb Z^n$ generated by the row vectors of $A$ . Now let $Ain M_rtimes n(mathbb Z)$ and $B:=PAQ$ where $Pin GL_rtimes r(mathbb Z), Q in GL_ntimes n(mathbb Z)$. Then how to prove that $mathbb Z^n/K_A $ and $mathbb Z^n/K_B$ are isomorphic as groups ?
I think the map $f: mathbb Z^n/K_A to mathbb Z^n/K_B$ defined as $f(bar a +K_A)=bar a Q+K_B, forall bar a in mathbb Z^n$ is an isomorphism, but I am not sure (I can't even show whether this map is well-defined or not).
Please help.
modules abelian-groups group-isomorphism free-modules free-abelian-group
asked Jul 28 at 17:11
user521337
606
That's basically it.
– Lord Shark the Unknown
Jul 28 at 17:13
@LordSharktheUnknown: Can you please at least provide a proof of the well-defined ness in an answer ?
– user521337
Jul 28 at 17:48
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider the free abelian group $mathbb Z^n$, with elements considered as row vectors. For every $Ain M_rtimes n(mathbb Z)$ , let $K_A$ be the subgroup of $mathbb Z^n$ generated by the row vectors of $A$ . Now let $Ain M_rtimes n(mathbb Z)$ and $B:=PAQ$ where $Pin GL_rtimes r(mathbb Z), Q in GL_ntimes n(mathbb Z)$. Then how to prove that $mathbb Z^n/K_A $ and $mathbb Z^n/K_B$ are isomorphic as groups ?
I think the map $f: mathbb Z^n/K_A to mathbb Z^n/K_B$ defined as $f(bar a +K_A)=bar a Q+K_B, forall bar a in mathbb Z^n$ is an isomorphism, but I am not sure (I can't even show whether this map is well-defined or not).
Please help.
modules abelian-groups group-isomorphism free-modules free-abelian-group
asked Jul 28 at 17:11
user521337
606
Consider the free abelian group $mathbb Z^n$, with elements considered as row vectors. For every $Ain M_rtimes n(mathbb Z)$ , let $K_A$ be the subgroup of $mathbb Z^n$ generated by the row vectors of $A$ . Now let $Ain M_rtimes n(mathbb Z)$ and $B:=PAQ$ where $Pin GL_rtimes r(mathbb Z), Q in GL_ntimes n(mathbb Z)$. Then how to prove that $mathbb Z^n/K_A $ and $mathbb Z^n/K_B$ are isomorphic as groups ?
I think the map $f: mathbb Z^n/K_A to mathbb Z^n/K_B$ defined as $f(bar a +K_A)=bar a Q+K_B, forall bar a in mathbb Z^n$ is an isomorphism, but I am not sure (I can't even show whether this map is well-defined or not).
Please help.
modules abelian-groups group-isomorphism free-modules free-abelian-group
asked Jul 28 at 17:11
user521337
606
asked Jul 28 at 17:11
user521337
606
asked Jul 28 at 17:11
user521337
606
asked Jul 28 at 17:11
user521337
606
606
That's basically it.
– Lord Shark the Unknown
Jul 28 at 17:13
@LordSharktheUnknown: Can you please at least provide a proof of the well-defined ness in an answer ?
– user521337
Jul 28 at 17:48
add a comment |Â
That's basically it.
– Lord Shark the Unknown
Jul 28 at 17:13
@LordSharktheUnknown: Can you please at least provide a proof of the well-defined ness in an answer ?
– user521337
Jul 28 at 17:48
That's basically it.
– Lord Shark the Unknown
Jul 28 at 17:13
That's basically it.
– Lord Shark the Unknown
Jul 28 at 17:13
@LordSharktheUnknown: Can you please at least provide a proof of the well-defined ness in an answer ?
– user521337
Jul 28 at 17:48
@LordSharktheUnknown: Can you please at least provide a proof of the well-defined ness in an answer ?
– user521337
Jul 28 at 17:48
add a comment |Â
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That's basically it.
– Lord Shark the Unknown
Jul 28 at 17:13
@LordSharktheUnknown: Can you please at least provide a proof of the well-defined ness in an answer ?
– user521337
Jul 28 at 17:48