Mixing
Ordering finite groups by sum of order of elements
Clash Royale CLAN TAG#URR8PPP
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Given two finite groups $G, H$, we are going to say that $G<_oH$ if either
a. $|G|<|H|$
or
b. $|G|=|H|$ and $displaystylesum_gin G o(g)<sum_hin H o(h)$,
where $o(g)$ denotes the order of the element $g$ (has this ordering a name?).
What is the smallest example (in this ordering) of a pair of nonisomorphic groups such that $G$ and $H$ are incomparable, i.e., such that they have same cardinal and same sum of orders of elements?
group-theory finite-groups order-theory
asked 1 hour ago
Jose Brox
2,4631921
add a comment |Â
up vote
6
down vote
favorite
Given two finite groups $G, H$, we are going to say that $G<_oH$ if either
a. $|G|<|H|$
or
b. $|G|=|H|$ and $displaystylesum_gin G o(g)<sum_hin H o(h)$,
where $o(g)$ denotes the order of the element $g$ (has this ordering a name?).
What is the smallest example (in this ordering) of a pair of nonisomorphic groups such that $G$ and $H$ are incomparable, i.e., such that they have same cardinal and same sum of orders of elements?
group-theory finite-groups order-theory
asked 1 hour ago
Jose Brox
2,4631921
1
Interesting question. Can you edit to show the best bound you have so far?
– Ethan Bolker
1 hour ago
1
@EthanBolker I don't have any computations yet. The thing is, I tend to think about "the smallest possible" group in some contexts, and I don't actually know if there is some simple "arithmetical" total ordering on finite groups in the sense of my question. Tomorrow I give a conference in which the concept of "smallest" group will have a minor role (in an informal way), and I've just thought that I cannot define the ordering, except if the one above does it (which I doubt). Of course, I can go with equivalence classes, but the doubt is now planted :)
– Jose Brox
1 hour ago
add a comment |Â
up vote
6
down vote
favorite
up vote
6
down vote
favorite
Given two finite groups $G, H$, we are going to say that $G<_oH$ if either
a. $|G|<|H|$
or
b. $|G|=|H|$ and $displaystylesum_gin G o(g)<sum_hin H o(h)$,
where $o(g)$ denotes the order of the element $g$ (has this ordering a name?).
What is the smallest example (in this ordering) of a pair of nonisomorphic groups such that $G$ and $H$ are incomparable, i.e., such that they have same cardinal and same sum of orders of elements?
group-theory finite-groups order-theory
asked 1 hour ago
Jose Brox
2,4631921
Given two finite groups $G, H$, we are going to say that $G<_oH$ if either
a. $|G|<|H|$
or
b. $|G|=|H|$ and $displaystylesum_gin G o(g)<sum_hin H o(h)$,
where $o(g)$ denotes the order of the element $g$ (has this ordering a name?).
What is the smallest example (in this ordering) of a pair of nonisomorphic groups such that $G$ and $H$ are incomparable, i.e., such that they have same cardinal and same sum of orders of elements?
group-theory finite-groups order-theory
group-theory finite-groups order-theory
asked 1 hour ago
Jose Brox
2,4631921
asked 1 hour ago
Jose Brox
2,4631921
asked 1 hour ago
Jose Brox
2,4631921
asked 1 hour ago
Jose Brox
2,4631921
asked 1 hour ago
Jose Brox
2,4631921
2,4631921
1
Interesting question. Can you edit to show the best bound you have so far?
– Ethan Bolker
1 hour ago
1
@EthanBolker I don't have any computations yet. The thing is, I tend to think about "the smallest possible" group in some contexts, and I don't actually know if there is some simple "arithmetical" total ordering on finite groups in the sense of my question. Tomorrow I give a conference in which the concept of "smallest" group will have a minor role (in an informal way), and I've just thought that I cannot define the ordering, except if the one above does it (which I doubt). Of course, I can go with equivalence classes, but the doubt is now planted :)
– Jose Brox
1 hour ago
add a comment |Â
1
Interesting question. Can you edit to show the best bound you have so far?
– Ethan Bolker
1 hour ago
1
@EthanBolker I don't have any computations yet. The thing is, I tend to think about "the smallest possible" group in some contexts, and I don't actually know if there is some simple "arithmetical" total ordering on finite groups in the sense of my question. Tomorrow I give a conference in which the concept of "smallest" group will have a minor role (in an informal way), and I've just thought that I cannot define the ordering, except if the one above does it (which I doubt). Of course, I can go with equivalence classes, but the doubt is now planted :)
– Jose Brox
1 hour ago
1
1
Interesting question. Can you edit to show the best bound you have so far?
– Ethan Bolker
1 hour ago
Interesting question. Can you edit to show the best bound you have so far?
– Ethan Bolker
1 hour ago
1
1
@EthanBolker I don't have any computations yet. The thing is, I tend to think about "the smallest possible" group in some contexts, and I don't actually know if there is some simple "arithmetical" total ordering on finite groups in the sense of my question. Tomorrow I give a conference in which the concept of "smallest" group will have a minor role (in an informal way), and I've just thought that I cannot define the ordering, except if the one above does it (which I doubt). Of course, I can go with equivalence classes, but the doubt is now planted :)
– Jose Brox
1 hour ago
@EthanBolker I don't have any computations yet. The thing is, I tend to think about "the smallest possible" group in some contexts, and I don't actually know if there is some simple "arithmetical" total ordering on finite groups in the sense of my question. Tomorrow I give a conference in which the concept of "smallest" group will have a minor role (in an informal way), and I've just thought that I cannot define the ordering, except if the one above does it (which I doubt). Of course, I can go with equivalence classes, but the doubt is now planted :)
– Jose Brox
1 hour ago
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
4
down vote
The following Maple function (code here requires the Maple package GroupTheory) takes as argument a group and returns the sum of the orders of its elements:
sumOfOrders := G -> add(u, u in map(PermOrder, convert(Elements(G), list), G));
This function takes as argument a positive integer $n$ and returns the multiset of sums of orders of elements of groups of order $n$:
sumOfOrdersList := n -> sort(map(sumOfOrders, AllSmallGroups(n)));
Checking the lowest orders manually we find that groups incomparable in your sense occur first in order 16, where there are three groups with sum $47$, three with sum $55$ and two with $87$.
(Cf. https://groupprops.subwiki.org/wiki/Groups_of_order_16 )
answered 25 mins ago
Travis
56.3k764138
add a comment |Â
up vote
3
down vote
Feed the following code to Magma(http://magma.maths.usyd.edu.au/calc/):
for grouporder in [1..24] do
printf "Checking sum of order for groups or order %o:", grouporder;
numgroup := NumberOfSmallGroups(grouporder);
sset := ;
for i in [1..numgroup] do
sumorder := 0;
G := SmallGroup(grouporder,i);
for j in G do
sumorder := sumorder + Order(j);
end for;
Append(~sset, sumorder);
end for;
sset;
printf "n";
end for;
You can see that there are two groups with "signature" $(16,47)$. Much thanks to Travis for pointing it out. Two of them are non-abelian; they have GAP id $(16,3)$ and $(16,13)$. Please refer the groupprop page for the description. The third is $mathbb Z/4mathbb Z times (mathbb Z/2mathbb Z)^2$. All has 1 element of order 1, 7 elements of order 2, and 8 elements of order 4.
Also, there are three groups with "signature" $(16,55)$.
The groups are the following:
$G_1 = (mathbb Z/4mathbb Z)^2$; $G_2 = langle a, b | a^4 = b^4 = 1, ba = ab^3rangle$ (the semi-direct product of two copies of $mathbb Z/4mathbb Z$), $G_3 = Q times mathbb Z/2mathbb Z$, where $Q$ is the quaternion group. All has 1 element of order 1, 3 elements of order 2, and 12 elements of order 4.
answered 22 mins ago
Hw Chu
2,845417
There are also three groups of order $16$ with order sum $47$, and in particular they are smaller with respect to the partial order $<_o$ than the groups mentioned here.
– Travis
18 mins ago
You're right. I did not see it as I skipped the sorting add looked at it by myself.
– Hw Chu
17 mins ago
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
The following Maple function (code here requires the Maple package GroupTheory) takes as argument a group and returns the sum of the orders of its elements:
sumOfOrders := G -> add(u, u in map(PermOrder, convert(Elements(G), list), G));
This function takes as argument a positive integer $n$ and returns the multiset of sums of orders of elements of groups of order $n$:
sumOfOrdersList := n -> sort(map(sumOfOrders, AllSmallGroups(n)));
Checking the lowest orders manually we find that groups incomparable in your sense occur first in order 16, where there are three groups with sum $47$, three with sum $55$ and two with $87$.
(Cf. https://groupprops.subwiki.org/wiki/Groups_of_order_16 )
answered 25 mins ago
Travis
56.3k764138
add a comment |Â
up vote
4
down vote
The following Maple function (code here requires the Maple package GroupTheory) takes as argument a group and returns the sum of the orders of its elements:
sumOfOrders := G -> add(u, u in map(PermOrder, convert(Elements(G), list), G));
This function takes as argument a positive integer $n$ and returns the multiset of sums of orders of elements of groups of order $n$:
sumOfOrdersList := n -> sort(map(sumOfOrders, AllSmallGroups(n)));
Checking the lowest orders manually we find that groups incomparable in your sense occur first in order 16, where there are three groups with sum $47$, three with sum $55$ and two with $87$.
(Cf. https://groupprops.subwiki.org/wiki/Groups_of_order_16 )
answered 25 mins ago
Travis
56.3k764138
add a comment |Â
up vote
4
down vote
up vote
4
down vote
The following Maple function (code here requires the Maple package GroupTheory) takes as argument a group and returns the sum of the orders of its elements:
sumOfOrders := G -> add(u, u in map(PermOrder, convert(Elements(G), list), G));
This function takes as argument a positive integer $n$ and returns the multiset of sums of orders of elements of groups of order $n$:
sumOfOrdersList := n -> sort(map(sumOfOrders, AllSmallGroups(n)));
Checking the lowest orders manually we find that groups incomparable in your sense occur first in order 16, where there are three groups with sum $47$, three with sum $55$ and two with $87$.
(Cf. https://groupprops.subwiki.org/wiki/Groups_of_order_16 )
answered 25 mins ago
Travis
56.3k764138
The following Maple function (code here requires the Maple package GroupTheory) takes as argument a group and returns the sum of the orders of its elements:
sumOfOrders := G -> add(u, u in map(PermOrder, convert(Elements(G), list), G));
This function takes as argument a positive integer $n$ and returns the multiset of sums of orders of elements of groups of order $n$:
sumOfOrdersList := n -> sort(map(sumOfOrders, AllSmallGroups(n)));
Checking the lowest orders manually we find that groups incomparable in your sense occur first in order 16, where there are three groups with sum $47$, three with sum $55$ and two with $87$.
(Cf. https://groupprops.subwiki.org/wiki/Groups_of_order_16 )
answered 25 mins ago
Travis
56.3k764138
answered 25 mins ago
Travis
56.3k764138
answered 25 mins ago
Travis
56.3k764138
answered 25 mins ago
Travis
56.3k764138
56.3k764138
add a comment |Â
add a comment |Â
up vote
3
down vote
Feed the following code to Magma(http://magma.maths.usyd.edu.au/calc/):
for grouporder in [1..24] do
printf "Checking sum of order for groups or order %o:", grouporder;
numgroup := NumberOfSmallGroups(grouporder);
sset := ;
for i in [1..numgroup] do
sumorder := 0;
G := SmallGroup(grouporder,i);
for j in G do
sumorder := sumorder + Order(j);
end for;
Append(~sset, sumorder);
end for;
sset;
printf "n";
end for;
You can see that there are two groups with "signature" $(16,47)$. Much thanks to Travis for pointing it out. Two of them are non-abelian; they have GAP id $(16,3)$ and $(16,13)$. Please refer the groupprop page for the description. The third is $mathbb Z/4mathbb Z times (mathbb Z/2mathbb Z)^2$. All has 1 element of order 1, 7 elements of order 2, and 8 elements of order 4.
Also, there are three groups with "signature" $(16,55)$.
The groups are the following:
$G_1 = (mathbb Z/4mathbb Z)^2$; $G_2 = langle a, b | a^4 = b^4 = 1, ba = ab^3rangle$ (the semi-direct product of two copies of $mathbb Z/4mathbb Z$), $G_3 = Q times mathbb Z/2mathbb Z$, where $Q$ is the quaternion group. All has 1 element of order 1, 3 elements of order 2, and 12 elements of order 4.
answered 22 mins ago
Hw Chu
2,845417
There are also three groups of order $16$ with order sum $47$, and in particular they are smaller with respect to the partial order $<_o$ than the groups mentioned here.
– Travis
18 mins ago
You're right. I did not see it as I skipped the sorting add looked at it by myself.
– Hw Chu
17 mins ago
add a comment |Â
up vote
3
down vote
Feed the following code to Magma(http://magma.maths.usyd.edu.au/calc/):
for grouporder in [1..24] do
printf "Checking sum of order for groups or order %o:", grouporder;
numgroup := NumberOfSmallGroups(grouporder);
sset := ;
for i in [1..numgroup] do
sumorder := 0;
G := SmallGroup(grouporder,i);
for j in G do
sumorder := sumorder + Order(j);
end for;
Append(~sset, sumorder);
end for;
sset;
printf "n";
end for;
You can see that there are two groups with "signature" $(16,47)$. Much thanks to Travis for pointing it out. Two of them are non-abelian; they have GAP id $(16,3)$ and $(16,13)$. Please refer the groupprop page for the description. The third is $mathbb Z/4mathbb Z times (mathbb Z/2mathbb Z)^2$. All has 1 element of order 1, 7 elements of order 2, and 8 elements of order 4.
Also, there are three groups with "signature" $(16,55)$.
The groups are the following:
$G_1 = (mathbb Z/4mathbb Z)^2$; $G_2 = langle a, b | a^4 = b^4 = 1, ba = ab^3rangle$ (the semi-direct product of two copies of $mathbb Z/4mathbb Z$), $G_3 = Q times mathbb Z/2mathbb Z$, where $Q$ is the quaternion group. All has 1 element of order 1, 3 elements of order 2, and 12 elements of order 4.
answered 22 mins ago
Hw Chu
2,845417
There are also three groups of order $16$ with order sum $47$, and in particular they are smaller with respect to the partial order $<_o$ than the groups mentioned here.
– Travis
18 mins ago
You're right. I did not see it as I skipped the sorting add looked at it by myself.
– Hw Chu
17 mins ago
add a comment |Â
up vote
3
down vote
up vote
3
down vote
Feed the following code to Magma(http://magma.maths.usyd.edu.au/calc/):
for grouporder in [1..24] do
printf "Checking sum of order for groups or order %o:", grouporder;
numgroup := NumberOfSmallGroups(grouporder);
sset := ;
for i in [1..numgroup] do
sumorder := 0;
G := SmallGroup(grouporder,i);
for j in G do
sumorder := sumorder + Order(j);
end for;
Append(~sset, sumorder);
end for;
sset;
printf "n";
end for;
You can see that there are two groups with "signature" $(16,47)$. Much thanks to Travis for pointing it out. Two of them are non-abelian; they have GAP id $(16,3)$ and $(16,13)$. Please refer the groupprop page for the description. The third is $mathbb Z/4mathbb Z times (mathbb Z/2mathbb Z)^2$. All has 1 element of order 1, 7 elements of order 2, and 8 elements of order 4.
Also, there are three groups with "signature" $(16,55)$.
The groups are the following:
$G_1 = (mathbb Z/4mathbb Z)^2$; $G_2 = langle a, b | a^4 = b^4 = 1, ba = ab^3rangle$ (the semi-direct product of two copies of $mathbb Z/4mathbb Z$), $G_3 = Q times mathbb Z/2mathbb Z$, where $Q$ is the quaternion group. All has 1 element of order 1, 3 elements of order 2, and 12 elements of order 4.
answered 22 mins ago
Hw Chu
2,845417
Feed the following code to Magma(http://magma.maths.usyd.edu.au/calc/):
for grouporder in [1..24] do
printf "Checking sum of order for groups or order %o:", grouporder;
numgroup := NumberOfSmallGroups(grouporder);
sset := ;
for i in [1..numgroup] do
sumorder := 0;
G := SmallGroup(grouporder,i);
for j in G do
sumorder := sumorder + Order(j);
end for;
Append(~sset, sumorder);
end for;
sset;
printf "n";
end for;
You can see that there are two groups with "signature" $(16,47)$. Much thanks to Travis for pointing it out. Two of them are non-abelian; they have GAP id $(16,3)$ and $(16,13)$. Please refer the groupprop page for the description. The third is $mathbb Z/4mathbb Z times (mathbb Z/2mathbb Z)^2$. All has 1 element of order 1, 7 elements of order 2, and 8 elements of order 4.
Also, there are three groups with "signature" $(16,55)$.
The groups are the following:
$G_1 = (mathbb Z/4mathbb Z)^2$; $G_2 = langle a, b | a^4 = b^4 = 1, ba = ab^3rangle$ (the semi-direct product of two copies of $mathbb Z/4mathbb Z$), $G_3 = Q times mathbb Z/2mathbb Z$, where $Q$ is the quaternion group. All has 1 element of order 1, 3 elements of order 2, and 12 elements of order 4.
answered 22 mins ago
Hw Chu
2,845417
edited 8 mins ago
answered 22 mins ago
Hw Chu
2,845417
answered 22 mins ago
Hw Chu
2,845417
answered 22 mins ago
Hw Chu
2,845417
2,845417
There are also three groups of order $16$ with order sum $47$, and in particular they are smaller with respect to the partial order $<_o$ than the groups mentioned here.
– Travis
18 mins ago
You're right. I did not see it as I skipped the sorting add looked at it by myself.
– Hw Chu
17 mins ago
add a comment |Â
There are also three groups of order $16$ with order sum $47$, and in particular they are smaller with respect to the partial order $<_o$ than the groups mentioned here.
– Travis
18 mins ago
You're right. I did not see it as I skipped the sorting add looked at it by myself.
– Hw Chu
17 mins ago
There are also three groups of order $16$ with order sum $47$, and in particular they are smaller with respect to the partial order $<_o$ than the groups mentioned here.
– Travis
18 mins ago
There are also three groups of order $16$ with order sum $47$, and in particular they are smaller with respect to the partial order $<_o$ than the groups mentioned here.
– Travis
18 mins ago
You're right. I did not see it as I skipped the sorting add looked at it by myself.
– Hw Chu
17 mins ago
You're right. I did not see it as I skipped the sorting add looked at it by myself.
– Hw Chu
17 mins ago
add a comment |Â
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1
Interesting question. Can you edit to show the best bound you have so far?
– Ethan Bolker
1 hour ago
1
@EthanBolker I don't have any computations yet. The thing is, I tend to think about "the smallest possible" group in some contexts, and I don't actually know if there is some simple "arithmetical" total ordering on finite groups in the sense of my question. Tomorrow I give a conference in which the concept of "smallest" group will have a minor role (in an informal way), and I've just thought that I cannot define the ordering, except if the one above does it (which I doubt). Of course, I can go with equivalence classes, but the doubt is now planted :)
– Jose Brox
1 hour ago