When group action on cartesian product is transitive?
$begingroup$
Let $G$ be a group acting transitively on sets $Omega$ and $Lambda$. Then there is a natural induced action of $G$ on cartesian product $Omega times Lambda$. I can prove that if $G$ is finite this action is transitive if and only if $G = G_{omega}G_{lambda}$ for all $omegainOmega, lambdainLambda$ (here $G_{omega}$ and $G_{lambda}$ are stabilisers of $omega$ and $lambda$, respectively).
My question. Is this true, when $G$ is infinite group?
abstract-algebra group-theory group-actions infinite-groups
asked Dec 22 '18 at 17:33
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
$endgroup$
add a comment |
$begingroup$
Let $G$ be a group acting transitively on sets $Omega$ and $Lambda$. Then there is a natural induced action of $G$ on cartesian product $Omega times Lambda$. I can prove that if $G$ is finite this action is transitive if and only if $G = G_{omega}G_{lambda}$ for all $omegainOmega, lambdainLambda$ (here $G_{omega}$ and $G_{lambda}$ are stabilisers of $omega$ and $lambda$, respectively).
My question. Is this true, when $G$ is infinite group?
abstract-algebra group-theory group-actions infinite-groups
asked Dec 22 '18 at 17:33
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
$endgroup$
add a comment |
$begingroup$
Let $G$ be a group acting transitively on sets $Omega$ and $Lambda$. Then there is a natural induced action of $G$ on cartesian product $Omega times Lambda$. I can prove that if $G$ is finite this action is transitive if and only if $G = G_{omega}G_{lambda}$ for all $omegainOmega, lambdainLambda$ (here $G_{omega}$ and $G_{lambda}$ are stabilisers of $omega$ and $lambda$, respectively).
My question. Is this true, when $G$ is infinite group?
abstract-algebra group-theory group-actions infinite-groups
asked Dec 22 '18 at 17:33
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
$endgroup$
Let $G$ be a group acting transitively on sets $Omega$ and $Lambda$. Then there is a natural induced action of $G$ on cartesian product $Omega times Lambda$. I can prove that if $G$ is finite this action is transitive if and only if $G = G_{omega}G_{lambda}$ for all $omegainOmega, lambdainLambda$ (here $G_{omega}$ and $G_{lambda}$ are stabilisers of $omega$ and $lambda$, respectively).
My question. Is this true, when $G$ is infinite group?
abstract-algebra group-theory group-actions infinite-groups
abstract-algebra group-theory group-actions infinite-groups
asked Dec 22 '18 at 17:33
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
asked Dec 22 '18 at 17:33
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
asked Dec 22 '18 at 17:33
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
asked Dec 22 '18 at 17:33
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
asked Dec 22 '18 at 17:33
Mikhail GoltvanitsaMikhail Goltvanitsa
623414
623414
add a comment |
add a comment |
1 Answer
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$begingroup$
Yes I believe so. This follows from two statenments.
If a group $G$ acts trantively on a set $X$, and $H le G$, then $H$ acts transitively on $X$ if and only if $G = HG_x$ for all $x in X$.
With your hypotheses, $G$ acts trasitively on $Omega times Lambda$ if and only if $G_omega$ acts transitively on $Lambda$ for all (or equivalently some) $omega in Omega$.
I don't really see how finiteness is involved in either of these claims. Which of these are you unsure of, and in which direction?
answered Dec 22 '18 at 18:21
Derek HoltDerek Holt
54.1k53571
$endgroup$
$begingroup$
thank you very much! Your first observation helped me to understand how to avoid arguments, which using finiteness.
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 18:52
add a comment |
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1 Answer
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1 Answer
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$begingroup$
Yes I believe so. This follows from two statenments.
If a group $G$ acts trantively on a set $X$, and $H le G$, then $H$ acts transitively on $X$ if and only if $G = HG_x$ for all $x in X$.
With your hypotheses, $G$ acts trasitively on $Omega times Lambda$ if and only if $G_omega$ acts transitively on $Lambda$ for all (or equivalently some) $omega in Omega$.
I don't really see how finiteness is involved in either of these claims. Which of these are you unsure of, and in which direction?
answered Dec 22 '18 at 18:21
Derek HoltDerek Holt
54.1k53571
$endgroup$
$begingroup$
thank you very much! Your first observation helped me to understand how to avoid arguments, which using finiteness.
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 18:52
add a comment |
$begingroup$
Yes I believe so. This follows from two statenments.
If a group $G$ acts trantively on a set $X$, and $H le G$, then $H$ acts transitively on $X$ if and only if $G = HG_x$ for all $x in X$.
With your hypotheses, $G$ acts trasitively on $Omega times Lambda$ if and only if $G_omega$ acts transitively on $Lambda$ for all (or equivalently some) $omega in Omega$.
I don't really see how finiteness is involved in either of these claims. Which of these are you unsure of, and in which direction?
answered Dec 22 '18 at 18:21
Derek HoltDerek Holt
54.1k53571
$endgroup$
$begingroup$
thank you very much! Your first observation helped me to understand how to avoid arguments, which using finiteness.
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 18:52
add a comment |
$begingroup$
Yes I believe so. This follows from two statenments.
If a group $G$ acts trantively on a set $X$, and $H le G$, then $H$ acts transitively on $X$ if and only if $G = HG_x$ for all $x in X$.
With your hypotheses, $G$ acts trasitively on $Omega times Lambda$ if and only if $G_omega$ acts transitively on $Lambda$ for all (or equivalently some) $omega in Omega$.
I don't really see how finiteness is involved in either of these claims. Which of these are you unsure of, and in which direction?
answered Dec 22 '18 at 18:21
Derek HoltDerek Holt
54.1k53571
$endgroup$
Yes I believe so. This follows from two statenments.
If a group $G$ acts trantively on a set $X$, and $H le G$, then $H$ acts transitively on $X$ if and only if $G = HG_x$ for all $x in X$.
With your hypotheses, $G$ acts trasitively on $Omega times Lambda$ if and only if $G_omega$ acts transitively on $Lambda$ for all (or equivalently some) $omega in Omega$.
I don't really see how finiteness is involved in either of these claims. Which of these are you unsure of, and in which direction?
answered Dec 22 '18 at 18:21
Derek HoltDerek Holt
54.1k53571
answered Dec 22 '18 at 18:21
Derek HoltDerek Holt
54.1k53571
answered Dec 22 '18 at 18:21
Derek HoltDerek Holt
54.1k53571
answered Dec 22 '18 at 18:21
Derek HoltDerek Holt
54.1k53571
54.1k53571
$begingroup$
thank you very much! Your first observation helped me to understand how to avoid arguments, which using finiteness.
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 18:52
add a comment |
$begingroup$
thank you very much! Your first observation helped me to understand how to avoid arguments, which using finiteness.
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 18:52
$begingroup$
thank you very much! Your first observation helped me to understand how to avoid arguments, which using finiteness.
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 18:52
$begingroup$
thank you very much! Your first observation helped me to understand how to avoid arguments, which using finiteness.
$endgroup$
– Mikhail Goltvanitsa
Dec 22 '18 at 18:52
add a comment |
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