Mixing
Does the hypergraph of subgroups determine a group?
Clash Royale CLAN TAG#URR8PPP
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A hypergraph is a pair $H=(V,E)$ where $Vneq emptyset$ is a set and $Esubseteqcal P(V)$ is a collection of subsets of $V$. We say two hypergraphs $H_i=(V_i, E_i)$ for $i=1,2$ are isomorphic if there is a bijection $f:V_1to V_2$ such that $f(e_1) in E_2$ for all $e_1in E_1$, and $f^-1(e_2) in E_1$ for all $e_2in E_2$.
If $G$ is a group, denote by $textSub(G)$ the collection of the subgroups of $G$.
Are there non-isomorphic groups $G,H$ such that the hypergraphs $(G, textSub(G))$ and $(H, textSub(H))$ are isomorphic?
co.combinatorics gr.group-theory graph-theory hypergraph
asked 4 hours ago
Dominic van der Zypen
13.1k43171
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up vote
3
down vote
favorite
A hypergraph is a pair $H=(V,E)$ where $Vneq emptyset$ is a set and $Esubseteqcal P(V)$ is a collection of subsets of $V$. We say two hypergraphs $H_i=(V_i, E_i)$ for $i=1,2$ are isomorphic if there is a bijection $f:V_1to V_2$ such that $f(e_1) in E_2$ for all $e_1in E_1$, and $f^-1(e_2) in E_1$ for all $e_2in E_2$.
If $G$ is a group, denote by $textSub(G)$ the collection of the subgroups of $G$.
Are there non-isomorphic groups $G,H$ such that the hypergraphs $(G, textSub(G))$ and $(H, textSub(H))$ are isomorphic?
co.combinatorics gr.group-theory graph-theory hypergraph
asked 4 hours ago
Dominic van der Zypen
13.1k43171
2
See mathoverflow.net/questions/37738/…
– Derek Holt
4 hours ago
What is the subgroup hypergraph of a cyclic group of prime order?
– Gerald Edgar
3 hours ago
So there is the important vertex which is characterized by being on every hyperedge.
– AHusain
1 hour ago
@GeraldEdgar If I understand you correctly, the answer is $(mathbbZ_p, big0,mathbbZ_pbig)$ for $p$ prime?
– Dominic van der Zypen
1 hour ago
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
A hypergraph is a pair $H=(V,E)$ where $Vneq emptyset$ is a set and $Esubseteqcal P(V)$ is a collection of subsets of $V$. We say two hypergraphs $H_i=(V_i, E_i)$ for $i=1,2$ are isomorphic if there is a bijection $f:V_1to V_2$ such that $f(e_1) in E_2$ for all $e_1in E_1$, and $f^-1(e_2) in E_1$ for all $e_2in E_2$.
If $G$ is a group, denote by $textSub(G)$ the collection of the subgroups of $G$.
Are there non-isomorphic groups $G,H$ such that the hypergraphs $(G, textSub(G))$ and $(H, textSub(H))$ are isomorphic?
co.combinatorics gr.group-theory graph-theory hypergraph
asked 4 hours ago
Dominic van der Zypen
13.1k43171
A hypergraph is a pair $H=(V,E)$ where $Vneq emptyset$ is a set and $Esubseteqcal P(V)$ is a collection of subsets of $V$. We say two hypergraphs $H_i=(V_i, E_i)$ for $i=1,2$ are isomorphic if there is a bijection $f:V_1to V_2$ such that $f(e_1) in E_2$ for all $e_1in E_1$, and $f^-1(e_2) in E_1$ for all $e_2in E_2$.
If $G$ is a group, denote by $textSub(G)$ the collection of the subgroups of $G$.
Are there non-isomorphic groups $G,H$ such that the hypergraphs $(G, textSub(G))$ and $(H, textSub(H))$ are isomorphic?
co.combinatorics gr.group-theory graph-theory hypergraph
co.combinatorics gr.group-theory graph-theory hypergraph
asked 4 hours ago
Dominic van der Zypen
13.1k43171
asked 4 hours ago
Dominic van der Zypen
13.1k43171
edited 1 hour ago
asked 4 hours ago
Dominic van der Zypen
13.1k43171
asked 4 hours ago
Dominic van der Zypen
13.1k43171
asked 4 hours ago
Dominic van der Zypen
13.1k43171
13.1k43171
2
See mathoverflow.net/questions/37738/…
– Derek Holt
4 hours ago
What is the subgroup hypergraph of a cyclic group of prime order?
– Gerald Edgar
3 hours ago
So there is the important vertex which is characterized by being on every hyperedge.
– AHusain
1 hour ago
@GeraldEdgar If I understand you correctly, the answer is $(mathbbZ_p, big0,mathbbZ_pbig)$ for $p$ prime?
– Dominic van der Zypen
1 hour ago
add a comment |Â
2
See mathoverflow.net/questions/37738/…
– Derek Holt
4 hours ago
What is the subgroup hypergraph of a cyclic group of prime order?
– Gerald Edgar
3 hours ago
So there is the important vertex which is characterized by being on every hyperedge.
– AHusain
1 hour ago
@GeraldEdgar If I understand you correctly, the answer is $(mathbbZ_p, big0,mathbbZ_pbig)$ for $p$ prime?
– Dominic van der Zypen
1 hour ago
2
2
See mathoverflow.net/questions/37738/…
– Derek Holt
4 hours ago
See mathoverflow.net/questions/37738/…
– Derek Holt
4 hours ago
What is the subgroup hypergraph of a cyclic group of prime order?
– Gerald Edgar
3 hours ago
What is the subgroup hypergraph of a cyclic group of prime order?
– Gerald Edgar
3 hours ago
So there is the important vertex which is characterized by being on every hyperedge.
– AHusain
1 hour ago
So there is the important vertex which is characterized by being on every hyperedge.
– AHusain
1 hour ago
@GeraldEdgar If I understand you correctly, the answer is $(mathbbZ_p, big0,mathbbZ_pbig)$ for $p$ prime?
– Dominic van der Zypen
1 hour ago
@GeraldEdgar If I understand you correctly, the answer is $(mathbbZ_p, big0,mathbbZ_pbig)$ for $p$ prime?
– Dominic van der Zypen
1 hour ago
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
4
down vote
accepted
In the comments to the question, I notice something which might be an error, or at least is an incomplete response. It is pointed out in the comments that there exist nonisomorphic groups with isomorphic subgroup lattices. While true, that fact doesn't answer this question, since it is possible to have isomorphic subgroup lattices and nonisomorphic subgroup hypergraphs. Even if you assume that the groups have the same order and isomorphic subgroup lattices, it does not immediately follow that they have isomorphic subgroup hypergraphs.
I am sure the negative answer to this question can be found in Roland Schmidt's book. But I would like to point to a theorem I coauthored after Schmidt's book was published, which applies to this question. Namely:
Thm. For any finite $N$, there is a finite set $X=X_N$ and $N$ binary operations $circ_i$ defined on $X$ such that
(1) $G_i = (X,circ_i)$ is a group for all $i$,
(2) $G_inotcong G_j$ when $ineq j$, and
(3) for all $i, j$, the groups
$G_i^kappa$ and $G_j^kappa$ have exactly the same subgroups (as sets) for all cardinals $kappa$.
The last item means that the subgroup hypergraphs of $G_i^kappa$ and $G_j^kappa$ are equal for all $i, j, kappa$.
The paper is
Keith A. Kearnes and Agnes Szendrei,
Groups with identical subgroup lattices in all powers.
J. Group Theory 7 (2004), no. 3, 385--402.
answered 45 mins ago
Keith Kearnes
5,31412438
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
In the comments to the question, I notice something which might be an error, or at least is an incomplete response. It is pointed out in the comments that there exist nonisomorphic groups with isomorphic subgroup lattices. While true, that fact doesn't answer this question, since it is possible to have isomorphic subgroup lattices and nonisomorphic subgroup hypergraphs. Even if you assume that the groups have the same order and isomorphic subgroup lattices, it does not immediately follow that they have isomorphic subgroup hypergraphs.
I am sure the negative answer to this question can be found in Roland Schmidt's book. But I would like to point to a theorem I coauthored after Schmidt's book was published, which applies to this question. Namely:
Thm. For any finite $N$, there is a finite set $X=X_N$ and $N$ binary operations $circ_i$ defined on $X$ such that
(1) $G_i = (X,circ_i)$ is a group for all $i$,
(2) $G_inotcong G_j$ when $ineq j$, and
(3) for all $i, j$, the groups
$G_i^kappa$ and $G_j^kappa$ have exactly the same subgroups (as sets) for all cardinals $kappa$.
The last item means that the subgroup hypergraphs of $G_i^kappa$ and $G_j^kappa$ are equal for all $i, j, kappa$.
The paper is
Keith A. Kearnes and Agnes Szendrei,
Groups with identical subgroup lattices in all powers.
J. Group Theory 7 (2004), no. 3, 385--402.
answered 45 mins ago
Keith Kearnes
5,31412438
add a comment |Â
up vote
4
down vote
accepted
In the comments to the question, I notice something which might be an error, or at least is an incomplete response. It is pointed out in the comments that there exist nonisomorphic groups with isomorphic subgroup lattices. While true, that fact doesn't answer this question, since it is possible to have isomorphic subgroup lattices and nonisomorphic subgroup hypergraphs. Even if you assume that the groups have the same order and isomorphic subgroup lattices, it does not immediately follow that they have isomorphic subgroup hypergraphs.
I am sure the negative answer to this question can be found in Roland Schmidt's book. But I would like to point to a theorem I coauthored after Schmidt's book was published, which applies to this question. Namely:
Thm. For any finite $N$, there is a finite set $X=X_N$ and $N$ binary operations $circ_i$ defined on $X$ such that
(1) $G_i = (X,circ_i)$ is a group for all $i$,
(2) $G_inotcong G_j$ when $ineq j$, and
(3) for all $i, j$, the groups
$G_i^kappa$ and $G_j^kappa$ have exactly the same subgroups (as sets) for all cardinals $kappa$.
The last item means that the subgroup hypergraphs of $G_i^kappa$ and $G_j^kappa$ are equal for all $i, j, kappa$.
The paper is
Keith A. Kearnes and Agnes Szendrei,
Groups with identical subgroup lattices in all powers.
J. Group Theory 7 (2004), no. 3, 385--402.
answered 45 mins ago
Keith Kearnes
5,31412438
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
In the comments to the question, I notice something which might be an error, or at least is an incomplete response. It is pointed out in the comments that there exist nonisomorphic groups with isomorphic subgroup lattices. While true, that fact doesn't answer this question, since it is possible to have isomorphic subgroup lattices and nonisomorphic subgroup hypergraphs. Even if you assume that the groups have the same order and isomorphic subgroup lattices, it does not immediately follow that they have isomorphic subgroup hypergraphs.
I am sure the negative answer to this question can be found in Roland Schmidt's book. But I would like to point to a theorem I coauthored after Schmidt's book was published, which applies to this question. Namely:
Thm. For any finite $N$, there is a finite set $X=X_N$ and $N$ binary operations $circ_i$ defined on $X$ such that
(1) $G_i = (X,circ_i)$ is a group for all $i$,
(2) $G_inotcong G_j$ when $ineq j$, and
(3) for all $i, j$, the groups
$G_i^kappa$ and $G_j^kappa$ have exactly the same subgroups (as sets) for all cardinals $kappa$.
The last item means that the subgroup hypergraphs of $G_i^kappa$ and $G_j^kappa$ are equal for all $i, j, kappa$.
The paper is
Keith A. Kearnes and Agnes Szendrei,
Groups with identical subgroup lattices in all powers.
J. Group Theory 7 (2004), no. 3, 385--402.
answered 45 mins ago
Keith Kearnes
5,31412438
In the comments to the question, I notice something which might be an error, or at least is an incomplete response. It is pointed out in the comments that there exist nonisomorphic groups with isomorphic subgroup lattices. While true, that fact doesn't answer this question, since it is possible to have isomorphic subgroup lattices and nonisomorphic subgroup hypergraphs. Even if you assume that the groups have the same order and isomorphic subgroup lattices, it does not immediately follow that they have isomorphic subgroup hypergraphs.
I am sure the negative answer to this question can be found in Roland Schmidt's book. But I would like to point to a theorem I coauthored after Schmidt's book was published, which applies to this question. Namely:
Thm. For any finite $N$, there is a finite set $X=X_N$ and $N$ binary operations $circ_i$ defined on $X$ such that
(1) $G_i = (X,circ_i)$ is a group for all $i$,
(2) $G_inotcong G_j$ when $ineq j$, and
(3) for all $i, j$, the groups
$G_i^kappa$ and $G_j^kappa$ have exactly the same subgroups (as sets) for all cardinals $kappa$.
The last item means that the subgroup hypergraphs of $G_i^kappa$ and $G_j^kappa$ are equal for all $i, j, kappa$.
The paper is
Keith A. Kearnes and Agnes Szendrei,
Groups with identical subgroup lattices in all powers.
J. Group Theory 7 (2004), no. 3, 385--402.
answered 45 mins ago
Keith Kearnes
5,31412438
edited 37 mins ago
answered 45 mins ago
Keith Kearnes
5,31412438
answered 45 mins ago
Keith Kearnes
5,31412438
answered 45 mins ago
Keith Kearnes
5,31412438
5,31412438
add a comment |Â
add a comment |Â
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2
See mathoverflow.net/questions/37738/…
– Derek Holt
4 hours ago
What is the subgroup hypergraph of a cyclic group of prime order?
– Gerald Edgar
3 hours ago
So there is the important vertex which is characterized by being on every hyperedge.
– AHusain
1 hour ago
@GeraldEdgar If I understand you correctly, the answer is $(mathbbZ_p, big0,mathbbZ_pbig)$ for $p$ prime?
– Dominic van der Zypen
1 hour ago