Are free groups and free actions related?
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Is there a connection between free groups and free actions, or is it that their names just happen to be the same? I'm studying groups theory at the moment, and haven't found any relation between the two, and I find it a bit odd that the names are so similar.
group-theory free-groups
asked Dec 25 '18 at 20:38
Rei HenigmanRei Henigman
717
$endgroup$
|
show 1 more comment
$begingroup$
Is there a connection between free groups and free actions, or is it that their names just happen to be the same? I'm studying groups theory at the moment, and haven't found any relation between the two, and I find it a bit odd that the names are so similar.
group-theory free-groups
asked Dec 25 '18 at 20:38
Rei HenigmanRei Henigman
717
$endgroup$
7
$begingroup$
Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
$endgroup$
– Qiaochu Yuan
Dec 25 '18 at 20:41
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"Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
$endgroup$
– Dietrich Burde
Dec 25 '18 at 21:08
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@QiaochuYuan that should be an answer.
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– freakish
Dec 25 '18 at 21:38
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@QiaochuYuan I'll take that as an answer, Thank you!
$endgroup$
– Rei Henigman
Dec 26 '18 at 19:12
1
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Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
$endgroup$
– Paul Plummer
Dec 27 '18 at 17:29
|
show 1 more comment
$begingroup$
Is there a connection between free groups and free actions, or is it that their names just happen to be the same? I'm studying groups theory at the moment, and haven't found any relation between the two, and I find it a bit odd that the names are so similar.
group-theory free-groups
asked Dec 25 '18 at 20:38
Rei HenigmanRei Henigman
717
$endgroup$
Is there a connection between free groups and free actions, or is it that their names just happen to be the same? I'm studying groups theory at the moment, and haven't found any relation between the two, and I find it a bit odd that the names are so similar.
group-theory free-groups
group-theory free-groups
asked Dec 25 '18 at 20:38
Rei HenigmanRei Henigman
717
asked Dec 25 '18 at 20:38
Rei HenigmanRei Henigman
717
asked Dec 25 '18 at 20:38
Rei HenigmanRei Henigman
717
asked Dec 25 '18 at 20:38
Rei HenigmanRei Henigman
717
asked Dec 25 '18 at 20:38
Rei HenigmanRei Henigman
717
717
7
$begingroup$
Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
$endgroup$
– Qiaochu Yuan
Dec 25 '18 at 20:41
$begingroup$
"Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
$endgroup$
– Dietrich Burde
Dec 25 '18 at 21:08
$begingroup$
@QiaochuYuan that should be an answer.
$endgroup$
– freakish
Dec 25 '18 at 21:38
$begingroup$
@QiaochuYuan I'll take that as an answer, Thank you!
$endgroup$
– Rei Henigman
Dec 26 '18 at 19:12
1
$begingroup$
Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
$endgroup$
– Paul Plummer
Dec 27 '18 at 17:29
|
show 1 more comment
7
$begingroup$
Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
$endgroup$
– Qiaochu Yuan
Dec 25 '18 at 20:41
$begingroup$
"Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
$endgroup$
– Dietrich Burde
Dec 25 '18 at 21:08
$begingroup$
@QiaochuYuan that should be an answer.
$endgroup$
– freakish
Dec 25 '18 at 21:38
$begingroup$
@QiaochuYuan I'll take that as an answer, Thank you!
$endgroup$
– Rei Henigman
Dec 26 '18 at 19:12
1
$begingroup$
Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
$endgroup$
– Paul Plummer
Dec 27 '18 at 17:29
7
7
$begingroup$
Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
$endgroup$
– Qiaochu Yuan
Dec 25 '18 at 20:41
$begingroup$
Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
$endgroup$
– Qiaochu Yuan
Dec 25 '18 at 20:41
$begingroup$
"Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
$endgroup$
– Dietrich Burde
Dec 25 '18 at 21:08
$begingroup$
"Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
$endgroup$
– Dietrich Burde
Dec 25 '18 at 21:08
$begingroup$
@QiaochuYuan that should be an answer.
$endgroup$
– freakish
Dec 25 '18 at 21:38
$begingroup$
@QiaochuYuan that should be an answer.
$endgroup$
– freakish
Dec 25 '18 at 21:38
$begingroup$
@QiaochuYuan I'll take that as an answer, Thank you!
$endgroup$
– Rei Henigman
Dec 26 '18 at 19:12
$begingroup$
@QiaochuYuan I'll take that as an answer, Thank you!
$endgroup$
– Rei Henigman
Dec 26 '18 at 19:12
1
1
$begingroup$
Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
$endgroup$
– Paul Plummer
Dec 27 '18 at 17:29
$begingroup$
Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
$endgroup$
– Paul Plummer
Dec 27 '18 at 17:29
|
show 1 more comment
1 Answer
1
active
oldest
votes
$begingroup$
Free $G$-actions are free objects in the category of G-sets.
More precisely, a free object on a set $I$ in this category is the same as a free $G$-set $X$ endowed with a map $Ito G$ meeting once each orbit; the obvious way to produce it is just considering $X=Gtimes I$ with action $gcdot (h,i)=(gh,i)$ and the map $Ito X$, $imapsto (1,i)$.
For a free action on a set $I$, the maps from various sets to $X$ satisfying the given property are the free generating families of $X$ as $G$-set.
$endgroup$
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$begingroup$
Free $G$-actions are free objects in the category of G-sets.
More precisely, a free object on a set $I$ in this category is the same as a free $G$-set $X$ endowed with a map $Ito G$ meeting once each orbit; the obvious way to produce it is just considering $X=Gtimes I$ with action $gcdot (h,i)=(gh,i)$ and the map $Ito X$, $imapsto (1,i)$.
For a free action on a set $I$, the maps from various sets to $X$ satisfying the given property are the free generating families of $X$ as $G$-set.
$endgroup$
add a comment |
$begingroup$
Free $G$-actions are free objects in the category of G-sets.
More precisely, a free object on a set $I$ in this category is the same as a free $G$-set $X$ endowed with a map $Ito G$ meeting once each orbit; the obvious way to produce it is just considering $X=Gtimes I$ with action $gcdot (h,i)=(gh,i)$ and the map $Ito X$, $imapsto (1,i)$.
For a free action on a set $I$, the maps from various sets to $X$ satisfying the given property are the free generating families of $X$ as $G$-set.
$endgroup$
add a comment |
$begingroup$
Free $G$-actions are free objects in the category of G-sets.
More precisely, a free object on a set $I$ in this category is the same as a free $G$-set $X$ endowed with a map $Ito G$ meeting once each orbit; the obvious way to produce it is just considering $X=Gtimes I$ with action $gcdot (h,i)=(gh,i)$ and the map $Ito X$, $imapsto (1,i)$.
For a free action on a set $I$, the maps from various sets to $X$ satisfying the given property are the free generating families of $X$ as $G$-set.
$endgroup$
Free $G$-actions are free objects in the category of G-sets.
More precisely, a free object on a set $I$ in this category is the same as a free $G$-set $X$ endowed with a map $Ito G$ meeting once each orbit; the obvious way to produce it is just considering $X=Gtimes I$ with action $gcdot (h,i)=(gh,i)$ and the map $Ito X$, $imapsto (1,i)$.
For a free action on a set $I$, the maps from various sets to $X$ satisfying the given property are the free generating families of $X$ as $G$-set.
answered Jan 11 at 3:09
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YCor
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7
$begingroup$
Free actions are free objects in the category of $G$-sets. This is probably not where the name comes from, though. They both mean free as in unconstrained.
$endgroup$
– Qiaochu Yuan
Dec 25 '18 at 20:41
$begingroup$
"Free" is a common word, like "torsion-free" etc. They are not necessarily all related.
$endgroup$
– Dietrich Burde
Dec 25 '18 at 21:08
$begingroup$
@QiaochuYuan that should be an answer.
$endgroup$
– freakish
Dec 25 '18 at 21:38
$begingroup$
@QiaochuYuan I'll take that as an answer, Thank you!
$endgroup$
– Rei Henigman
Dec 26 '18 at 19:12
1
$begingroup$
Free groups are exactly the groups which act freely on trees(of course this is a very special case of a free action).
$endgroup$
– Paul Plummer
Dec 27 '18 at 17:29