Set of Double Cosets
Consider a group $G$ with subgroups $H$ and $K$. The double cosets ${ HgK : g in G }$ partition $G$. It seems this is just a set, not a group action. But I thought I might be missing something- can this be viewed as a group action of some group? I suppose it is the set of $K$-orbits of the group action of $G$ on $G/H$ (and vice versa). Also, I thought perhaps that $G$ could act, where for $g, g_0 in G$, $g cdot Hg_0 K = H g g_0 K$, but it seems this is not well defined.
More generally, I would appreciate any answer stating whether there is a universal property of this construction in some category.
group-theory group-actions
asked Nov 25 at 15:30
Dean Young
1,464720
add a comment |
Consider a group $G$ with subgroups $H$ and $K$. The double cosets ${ HgK : g in G }$ partition $G$. It seems this is just a set, not a group action. But I thought I might be missing something- can this be viewed as a group action of some group? I suppose it is the set of $K$-orbits of the group action of $G$ on $G/H$ (and vice versa). Also, I thought perhaps that $G$ could act, where for $g, g_0 in G$, $g cdot Hg_0 K = H g g_0 K$, but it seems this is not well defined.
More generally, I would appreciate any answer stating whether there is a universal property of this construction in some category.
group-theory group-actions
asked Nov 25 at 15:30
Dean Young
1,464720
add a comment |
Consider a group $G$ with subgroups $H$ and $K$. The double cosets ${ HgK : g in G }$ partition $G$. It seems this is just a set, not a group action. But I thought I might be missing something- can this be viewed as a group action of some group? I suppose it is the set of $K$-orbits of the group action of $G$ on $G/H$ (and vice versa). Also, I thought perhaps that $G$ could act, where for $g, g_0 in G$, $g cdot Hg_0 K = H g g_0 K$, but it seems this is not well defined.
More generally, I would appreciate any answer stating whether there is a universal property of this construction in some category.
group-theory group-actions
asked Nov 25 at 15:30
Dean Young
1,464720
Consider a group $G$ with subgroups $H$ and $K$. The double cosets ${ HgK : g in G }$ partition $G$. It seems this is just a set, not a group action. But I thought I might be missing something- can this be viewed as a group action of some group? I suppose it is the set of $K$-orbits of the group action of $G$ on $G/H$ (and vice versa). Also, I thought perhaps that $G$ could act, where for $g, g_0 in G$, $g cdot Hg_0 K = H g g_0 K$, but it seems this is not well defined.
More generally, I would appreciate any answer stating whether there is a universal property of this construction in some category.
group-theory group-actions
group-theory group-actions
asked Nov 25 at 15:30
Dean Young
1,464720
asked Nov 25 at 15:30
Dean Young
1,464720
edited Nov 25 at 15:46
asked Nov 25 at 15:30
Dean Young
1,464720
asked Nov 25 at 15:30
Dean Young
1,464720
asked Nov 25 at 15:30
Dean Young
1,464720
1,464720
add a comment |
add a comment |
1 Answer
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If I understand your first question correctly, then the answer is that these double cosets are the orbits of the (right) action of the direct product $H times K$ on $G$ given by
$$
g^{(h, k)} = h^{-1} g k.
$$
(I denote the action by an exponent.)
answered Nov 25 at 16:23
Andreas Caranti
55.9k34295
Or the left action given by $(h,k)g:=hgk^{-1}$.
– anon
Dec 3 at 2:01
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
If I understand your first question correctly, then the answer is that these double cosets are the orbits of the (right) action of the direct product $H times K$ on $G$ given by
$$
g^{(h, k)} = h^{-1} g k.
$$
(I denote the action by an exponent.)
answered Nov 25 at 16:23
Andreas Caranti
55.9k34295
Or the left action given by $(h,k)g:=hgk^{-1}$.
– anon
Dec 3 at 2:01
add a comment |
If I understand your first question correctly, then the answer is that these double cosets are the orbits of the (right) action of the direct product $H times K$ on $G$ given by
$$
g^{(h, k)} = h^{-1} g k.
$$
(I denote the action by an exponent.)
answered Nov 25 at 16:23
Andreas Caranti
55.9k34295
Or the left action given by $(h,k)g:=hgk^{-1}$.
– anon
Dec 3 at 2:01
add a comment |
If I understand your first question correctly, then the answer is that these double cosets are the orbits of the (right) action of the direct product $H times K$ on $G$ given by
$$
g^{(h, k)} = h^{-1} g k.
$$
(I denote the action by an exponent.)
answered Nov 25 at 16:23
Andreas Caranti
55.9k34295
If I understand your first question correctly, then the answer is that these double cosets are the orbits of the (right) action of the direct product $H times K$ on $G$ given by
$$
g^{(h, k)} = h^{-1} g k.
$$
(I denote the action by an exponent.)
answered Nov 25 at 16:23
Andreas Caranti
55.9k34295
answered Nov 25 at 16:23
Andreas Caranti
55.9k34295
answered Nov 25 at 16:23
Andreas Caranti
55.9k34295
answered Nov 25 at 16:23
Andreas Caranti
55.9k34295
55.9k34295
Or the left action given by $(h,k)g:=hgk^{-1}$.
– anon
Dec 3 at 2:01
add a comment |
Or the left action given by $(h,k)g:=hgk^{-1}$.
– anon
Dec 3 at 2:01
Or the left action given by $(h,k)g:=hgk^{-1}$.
– anon
Dec 3 at 2:01
Or the left action given by $(h,k)g:=hgk^{-1}$.
– anon
Dec 3 at 2:01
add a comment |
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