Subgroup of the direct product of groups
Consider a group $Gtimes H$ (The direct product of groups). I came across with the following theorem (Link):
if $A$ is a subgroup of $G$ and $B$ is a subgroup of $H$, then the direct product $Atimes B$ is a subgroup of $G times H$.
It made me wonder if $(Atimes B) leq (Gtimes H)$ then $Aleq G$ and $Bleq H$? Tried to find an example which disproves it but they all did not. Is the theorem valid? If so, how should I prove it?
EDIT:
I would like to find two groups $A$ and $B$ so $Atimes B leq Gtimes H$ but $Anot leq G$ or $Bnotleq H$ (or both).
group-theory
asked Dec 1 '18 at 13:53
vesii
685
|
show 7 more comments
Consider a group $Gtimes H$ (The direct product of groups). I came across with the following theorem (Link):
if $A$ is a subgroup of $G$ and $B$ is a subgroup of $H$, then the direct product $Atimes B$ is a subgroup of $G times H$.
It made me wonder if $(Atimes B) leq (Gtimes H)$ then $Aleq G$ and $Bleq H$? Tried to find an example which disproves it but they all did not. Is the theorem valid? If so, how should I prove it?
EDIT:
I would like to find two groups $A$ and $B$ so $Atimes B leq Gtimes H$ but $Anot leq G$ or $Bnotleq H$ (or both).
group-theory
asked Dec 1 '18 at 13:53
vesii
685
1
It depends what you mean by $A times B le G times H$. The notation you use seems to assume that $A le G$ and $B le H$, in which case what you ask is tautologically true.
– Derek Holt
Dec 1 '18 at 17:12
$Atimes B leq G times H$ means $Atimes B$ is a subgroup of $Gtimes H$
– vesii
Dec 1 '18 at 17:23
1
Yes but not all subgroups of $G times H$ can be written in the form $A times B$. So why are you writing the subgroup as $A times B$?
– Derek Holt
Dec 1 '18 at 17:53
@DerekHolt I think that we don't understand each other. Of course not all of the subgroups of $Gtimes H$ can be written as $Atimes B$. I want to find two groups $A$ and $B$ so $Atimes B leq Gtimes H$ but $Anot leq G$ or $Bnotleq H$ (or both).
– vesii
Dec 1 '18 at 20:13
Think about the notation you are using. It's wrong.
– the_fox
Dec 1 '18 at 20:21
|
show 7 more comments
Consider a group $Gtimes H$ (The direct product of groups). I came across with the following theorem (Link):
if $A$ is a subgroup of $G$ and $B$ is a subgroup of $H$, then the direct product $Atimes B$ is a subgroup of $G times H$.
It made me wonder if $(Atimes B) leq (Gtimes H)$ then $Aleq G$ and $Bleq H$? Tried to find an example which disproves it but they all did not. Is the theorem valid? If so, how should I prove it?
EDIT:
I would like to find two groups $A$ and $B$ so $Atimes B leq Gtimes H$ but $Anot leq G$ or $Bnotleq H$ (or both).
group-theory
asked Dec 1 '18 at 13:53
vesii
685
Consider a group $Gtimes H$ (The direct product of groups). I came across with the following theorem (Link):
if $A$ is a subgroup of $G$ and $B$ is a subgroup of $H$, then the direct product $Atimes B$ is a subgroup of $G times H$.
It made me wonder if $(Atimes B) leq (Gtimes H)$ then $Aleq G$ and $Bleq H$? Tried to find an example which disproves it but they all did not. Is the theorem valid? If so, how should I prove it?
EDIT:
I would like to find two groups $A$ and $B$ so $Atimes B leq Gtimes H$ but $Anot leq G$ or $Bnotleq H$ (or both).
group-theory
group-theory
asked Dec 1 '18 at 13:53
vesii
685
asked Dec 1 '18 at 13:53
vesii
685
edited Dec 1 '18 at 20:13
asked Dec 1 '18 at 13:53
vesii
685
asked Dec 1 '18 at 13:53
vesii
685
asked Dec 1 '18 at 13:53
vesii
685
685
1
It depends what you mean by $A times B le G times H$. The notation you use seems to assume that $A le G$ and $B le H$, in which case what you ask is tautologically true.
– Derek Holt
Dec 1 '18 at 17:12
$Atimes B leq G times H$ means $Atimes B$ is a subgroup of $Gtimes H$
– vesii
Dec 1 '18 at 17:23
1
Yes but not all subgroups of $G times H$ can be written in the form $A times B$. So why are you writing the subgroup as $A times B$?
– Derek Holt
Dec 1 '18 at 17:53
@DerekHolt I think that we don't understand each other. Of course not all of the subgroups of $Gtimes H$ can be written as $Atimes B$. I want to find two groups $A$ and $B$ so $Atimes B leq Gtimes H$ but $Anot leq G$ or $Bnotleq H$ (or both).
– vesii
Dec 1 '18 at 20:13
Think about the notation you are using. It's wrong.
– the_fox
Dec 1 '18 at 20:21
|
show 7 more comments
1
It depends what you mean by $A times B le G times H$. The notation you use seems to assume that $A le G$ and $B le H$, in which case what you ask is tautologically true.
– Derek Holt
Dec 1 '18 at 17:12
$Atimes B leq G times H$ means $Atimes B$ is a subgroup of $Gtimes H$
– vesii
Dec 1 '18 at 17:23
1
Yes but not all subgroups of $G times H$ can be written in the form $A times B$. So why are you writing the subgroup as $A times B$?
– Derek Holt
Dec 1 '18 at 17:53
@DerekHolt I think that we don't understand each other. Of course not all of the subgroups of $Gtimes H$ can be written as $Atimes B$. I want to find two groups $A$ and $B$ so $Atimes B leq Gtimes H$ but $Anot leq G$ or $Bnotleq H$ (or both).
– vesii
Dec 1 '18 at 20:13
Think about the notation you are using. It's wrong.
– the_fox
Dec 1 '18 at 20:21
1
1
It depends what you mean by $A times B le G times H$. The notation you use seems to assume that $A le G$ and $B le H$, in which case what you ask is tautologically true.
– Derek Holt
Dec 1 '18 at 17:12
It depends what you mean by $A times B le G times H$. The notation you use seems to assume that $A le G$ and $B le H$, in which case what you ask is tautologically true.
– Derek Holt
Dec 1 '18 at 17:12
$Atimes B leq G times H$ means $Atimes B$ is a subgroup of $Gtimes H$
– vesii
Dec 1 '18 at 17:23
$Atimes B leq G times H$ means $Atimes B$ is a subgroup of $Gtimes H$
– vesii
Dec 1 '18 at 17:23
1
1
Yes but not all subgroups of $G times H$ can be written in the form $A times B$. So why are you writing the subgroup as $A times B$?
– Derek Holt
Dec 1 '18 at 17:53
Yes but not all subgroups of $G times H$ can be written in the form $A times B$. So why are you writing the subgroup as $A times B$?
– Derek Holt
Dec 1 '18 at 17:53
@DerekHolt I think that we don't understand each other. Of course not all of the subgroups of $Gtimes H$ can be written as $Atimes B$. I want to find two groups $A$ and $B$ so $Atimes B leq Gtimes H$ but $Anot leq G$ or $Bnotleq H$ (or both).
– vesii
Dec 1 '18 at 20:13
@DerekHolt I think that we don't understand each other. Of course not all of the subgroups of $Gtimes H$ can be written as $Atimes B$. I want to find two groups $A$ and $B$ so $Atimes B leq Gtimes H$ but $Anot leq G$ or $Bnotleq H$ (or both).
– vesii
Dec 1 '18 at 20:13
Think about the notation you are using. It's wrong.
– the_fox
Dec 1 '18 at 20:21
Think about the notation you are using. It's wrong.
– the_fox
Dec 1 '18 at 20:21
|
show 7 more comments
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1
It depends what you mean by $A times B le G times H$. The notation you use seems to assume that $A le G$ and $B le H$, in which case what you ask is tautologically true.
– Derek Holt
Dec 1 '18 at 17:12
$Atimes B leq G times H$ means $Atimes B$ is a subgroup of $Gtimes H$
– vesii
Dec 1 '18 at 17:23
1
Yes but not all subgroups of $G times H$ can be written in the form $A times B$. So why are you writing the subgroup as $A times B$?
– Derek Holt
Dec 1 '18 at 17:53
@DerekHolt I think that we don't understand each other. Of course not all of the subgroups of $Gtimes H$ can be written as $Atimes B$. I want to find two groups $A$ and $B$ so $Atimes B leq Gtimes H$ but $Anot leq G$ or $Bnotleq H$ (or both).
– vesii
Dec 1 '18 at 20:13
Think about the notation you are using. It's wrong.
– the_fox
Dec 1 '18 at 20:21