abelian group as Z moduleabelian and finite groupModule, vector space and abelian groupDeterming whether (S, *) is an Abelian groupLocally graded group with all proper subgroups abelianHow is an abelian $G$-operator group with $m1 = m$ a $mathbb Z[G]$-moduleThe homomorphic image of an abelian group is abelianProve that middle cancellation implies that the group is abelianIs $G$ an Abelian group?Prove or disapprove: Group of order $135$ must be abelian“Module” versus “Abelian Group” of the Integers, German Language usage circa 1962
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abelian group as Z module
abelian and finite groupModule, vector space and abelian groupDeterming whether (S, *) is an Abelian groupLocally graded group with all proper subgroups abelianHow is an abelian $G$-operator group with $m1 = m$ a $mathbb Z[G]$-moduleThe homomorphic image of an abelian group is abelianProve that middle cancellation implies that the group is abelianIs $G$ an Abelian group?Prove or disapprove: Group of order $135$ must be abelian“Module” versus “Abelian Group” of the Integers, German Language usage circa 1962
$begingroup$
How Would you prove that every abelian group can be understood as a Z-Module in a unique way?
I would guess that you would have to prove its bijective, but not sure how to go about this
linear-algebra group-theory abelian-groups
asked Nov 4 '14 at 13:25
user189820user189820
1
$endgroup$
add a comment |
$begingroup$
How Would you prove that every abelian group can be understood as a Z-Module in a unique way?
I would guess that you would have to prove its bijective, but not sure how to go about this
linear-algebra group-theory abelian-groups
asked Nov 4 '14 at 13:25
user189820user189820
1
$endgroup$
1
$begingroup$
What is the definition of a $mathbb Z$-module?
$endgroup$
– awllower
Nov 4 '14 at 13:30
add a comment |
$begingroup$
How Would you prove that every abelian group can be understood as a Z-Module in a unique way?
I would guess that you would have to prove its bijective, but not sure how to go about this
linear-algebra group-theory abelian-groups
asked Nov 4 '14 at 13:25
user189820user189820
1
$endgroup$
How Would you prove that every abelian group can be understood as a Z-Module in a unique way?
I would guess that you would have to prove its bijective, but not sure how to go about this
linear-algebra group-theory abelian-groups
linear-algebra group-theory abelian-groups
asked Nov 4 '14 at 13:25
user189820user189820
1
asked Nov 4 '14 at 13:25
user189820user189820
1
asked Nov 4 '14 at 13:25
user189820user189820
1
asked Nov 4 '14 at 13:25
user189820user189820
1
asked Nov 4 '14 at 13:25
user189820user189820
1
1
1
$begingroup$
What is the definition of a $mathbb Z$-module?
$endgroup$
– awllower
Nov 4 '14 at 13:30
add a comment |
1
$begingroup$
What is the definition of a $mathbb Z$-module?
$endgroup$
– awllower
Nov 4 '14 at 13:30
1
1
$begingroup$
What is the definition of a $mathbb Z$-module?
$endgroup$
– awllower
Nov 4 '14 at 13:30
$begingroup$
What is the definition of a $mathbb Z$-module?
$endgroup$
– awllower
Nov 4 '14 at 13:30
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Given an abelian group $G$ its set of endomorphisms
$$
rm End(G)=f:Grightarrow G,texthomomorphism
$$
is a ring under the usual operations of sum and composition, with unity the identity map.
Given any ring with unity $R$ there is a unique map of unital rings
$$
Bbb Zlongrightarrow R
$$
given by $nmapsto ncdot1_R$ with the usual convention for $n<0$ (preserving the identity mandates $1_Bbb Zmapsto 1_R$). By applying this to $rm End(G)$ you immediately see that $G$ has a unique structure of $Bbb Z$-module.
edited Mar 29 at 10:02
Shlomi A
264213
answered Nov 4 '14 at 13:48
Andrea MoriAndrea Mori
20.1k13466
$endgroup$
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Given an abelian group $G$ its set of endomorphisms
$$
rm End(G)=f:Grightarrow G,texthomomorphism
$$
is a ring under the usual operations of sum and composition, with unity the identity map.
Given any ring with unity $R$ there is a unique map of unital rings
$$
Bbb Zlongrightarrow R
$$
given by $nmapsto ncdot1_R$ with the usual convention for $n<0$ (preserving the identity mandates $1_Bbb Zmapsto 1_R$). By applying this to $rm End(G)$ you immediately see that $G$ has a unique structure of $Bbb Z$-module.
edited Mar 29 at 10:02
Shlomi A
264213
answered Nov 4 '14 at 13:48
Andrea MoriAndrea Mori
20.1k13466
$endgroup$
add a comment |
$begingroup$
Given an abelian group $G$ its set of endomorphisms
$$
rm End(G)=f:Grightarrow G,texthomomorphism
$$
is a ring under the usual operations of sum and composition, with unity the identity map.
Given any ring with unity $R$ there is a unique map of unital rings
$$
Bbb Zlongrightarrow R
$$
given by $nmapsto ncdot1_R$ with the usual convention for $n<0$ (preserving the identity mandates $1_Bbb Zmapsto 1_R$). By applying this to $rm End(G)$ you immediately see that $G$ has a unique structure of $Bbb Z$-module.
edited Mar 29 at 10:02
Shlomi A
264213
answered Nov 4 '14 at 13:48
Andrea MoriAndrea Mori
20.1k13466
$endgroup$
add a comment |
$begingroup$
Given an abelian group $G$ its set of endomorphisms
$$
rm End(G)=f:Grightarrow G,texthomomorphism
$$
is a ring under the usual operations of sum and composition, with unity the identity map.
Given any ring with unity $R$ there is a unique map of unital rings
$$
Bbb Zlongrightarrow R
$$
given by $nmapsto ncdot1_R$ with the usual convention for $n<0$ (preserving the identity mandates $1_Bbb Zmapsto 1_R$). By applying this to $rm End(G)$ you immediately see that $G$ has a unique structure of $Bbb Z$-module.
edited Mar 29 at 10:02
Shlomi A
264213
answered Nov 4 '14 at 13:48
Andrea MoriAndrea Mori
20.1k13466
$endgroup$
Given an abelian group $G$ its set of endomorphisms
$$
rm End(G)=f:Grightarrow G,texthomomorphism
$$
is a ring under the usual operations of sum and composition, with unity the identity map.
Given any ring with unity $R$ there is a unique map of unital rings
$$
Bbb Zlongrightarrow R
$$
given by $nmapsto ncdot1_R$ with the usual convention for $n<0$ (preserving the identity mandates $1_Bbb Zmapsto 1_R$). By applying this to $rm End(G)$ you immediately see that $G$ has a unique structure of $Bbb Z$-module.
edited Mar 29 at 10:02
Shlomi A
264213
answered Nov 4 '14 at 13:48
Andrea MoriAndrea Mori
20.1k13466
edited Mar 29 at 10:02
Shlomi A
264213
edited Mar 29 at 10:02
Shlomi A
264213
edited Mar 29 at 10:02
Shlomi A
264213
264213
answered Nov 4 '14 at 13:48
Andrea MoriAndrea Mori
20.1k13466
answered Nov 4 '14 at 13:48
Andrea MoriAndrea Mori
20.1k13466
answered Nov 4 '14 at 13:48
Andrea MoriAndrea Mori
20.1k13466
20.1k13466
add a comment |
add a comment |
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1
$begingroup$
What is the definition of a $mathbb Z$-module?
$endgroup$
– awllower
Nov 4 '14 at 13:30