Mixing
How many Groups there are on a finite set?
$begingroup$
Let say cardinality of set S is $n=|S|$.
We know that there are $n^{n^2}$ all binary operations on that set.
To find out how many groups can be created by this set and by those operations, we need not only to know how many associative operations there are on that finite set.
But also this set and given operation must satisfy specific axioms: closure, associativity, identity and invertibility.
So how find out how many different groups can be created on that finite countable set?
abstract-algebra group-theory finite-groups binary-operations groups-enumeration
edited Jan 28 at 2:06
Zuriel
1,8881228
asked Dec 6 '13 at 6:44
IremadzeArchil19910311IremadzeArchil19910311
71021025
$endgroup$
|
show 3 more comments
$begingroup$
Let say cardinality of set S is $n=|S|$.
We know that there are $n^{n^2}$ all binary operations on that set.
To find out how many groups can be created by this set and by those operations, we need not only to know how many associative operations there are on that finite set.
But also this set and given operation must satisfy specific axioms: closure, associativity, identity and invertibility.
So how find out how many different groups can be created on that finite countable set?
abstract-algebra group-theory finite-groups binary-operations groups-enumeration
edited Jan 28 at 2:06
Zuriel
1,8881228
asked Dec 6 '13 at 6:44
IremadzeArchil19910311IremadzeArchil19910311
71021025
$endgroup$
$begingroup$
What do you want to know exactly? This is a very difficult question in general, though it's possible to answer in specific cases using theorems in group theory. For example, when $n$ is prime, there are $ncdot (n-2)!$ possible groups. (This follows from the fact that every group of prime order is cyclic, as well as some counting.)
$endgroup$
– Jim Belk
Dec 6 '13 at 6:51
$begingroup$
I need to know most likely in general case!
$endgroup$
– IremadzeArchil19910311
Dec 6 '13 at 6:52
2
$begingroup$
There is an OEIS entry for the sequence that you're looking for. Starting with $n=1$, the sequence is 1, 2, 3, 16, 30, 480, 840, 22080, 68040, 1088640, 3991680, and so forth.
$endgroup$
– Jim Belk
Dec 6 '13 at 7:00
2
$begingroup$
But usually, different groups mean non-isomorphic groups.
$endgroup$
– Derek Holt
Dec 6 '13 at 8:49
1
$begingroup$
@DerekHolt The number of non-isomorphic groups of order $n$ is the very first sequence in OEIS! oeis.org/…
$endgroup$
– MartianInvader
Jul 11 '14 at 22:27
|
show 3 more comments
$begingroup$
Let say cardinality of set S is $n=|S|$.
We know that there are $n^{n^2}$ all binary operations on that set.
To find out how many groups can be created by this set and by those operations, we need not only to know how many associative operations there are on that finite set.
But also this set and given operation must satisfy specific axioms: closure, associativity, identity and invertibility.
So how find out how many different groups can be created on that finite countable set?
abstract-algebra group-theory finite-groups binary-operations groups-enumeration
edited Jan 28 at 2:06
Zuriel
1,8881228
asked Dec 6 '13 at 6:44
IremadzeArchil19910311IremadzeArchil19910311
71021025
$endgroup$
Let say cardinality of set S is $n=|S|$.
We know that there are $n^{n^2}$ all binary operations on that set.
To find out how many groups can be created by this set and by those operations, we need not only to know how many associative operations there are on that finite set.
But also this set and given operation must satisfy specific axioms: closure, associativity, identity and invertibility.
So how find out how many different groups can be created on that finite countable set?
abstract-algebra group-theory finite-groups binary-operations groups-enumeration
abstract-algebra group-theory finite-groups binary-operations groups-enumeration
edited Jan 28 at 2:06
Zuriel
1,8881228
asked Dec 6 '13 at 6:44
IremadzeArchil19910311IremadzeArchil19910311
71021025
edited Jan 28 at 2:06
Zuriel
1,8881228
asked Dec 6 '13 at 6:44
IremadzeArchil19910311IremadzeArchil19910311
71021025
edited Jan 28 at 2:06
Zuriel
1,8881228
edited Jan 28 at 2:06
Zuriel
1,8881228
edited Jan 28 at 2:06
Zuriel
1,8881228
1,8881228
asked Dec 6 '13 at 6:44
IremadzeArchil19910311IremadzeArchil19910311
71021025
asked Dec 6 '13 at 6:44
IremadzeArchil19910311IremadzeArchil19910311
71021025
asked Dec 6 '13 at 6:44
IremadzeArchil19910311IremadzeArchil19910311
71021025
71021025
$begingroup$
What do you want to know exactly? This is a very difficult question in general, though it's possible to answer in specific cases using theorems in group theory. For example, when $n$ is prime, there are $ncdot (n-2)!$ possible groups. (This follows from the fact that every group of prime order is cyclic, as well as some counting.)
$endgroup$
– Jim Belk
Dec 6 '13 at 6:51
$begingroup$
I need to know most likely in general case!
$endgroup$
– IremadzeArchil19910311
Dec 6 '13 at 6:52
2
$begingroup$
There is an OEIS entry for the sequence that you're looking for. Starting with $n=1$, the sequence is 1, 2, 3, 16, 30, 480, 840, 22080, 68040, 1088640, 3991680, and so forth.
$endgroup$
– Jim Belk
Dec 6 '13 at 7:00
2
$begingroup$
But usually, different groups mean non-isomorphic groups.
$endgroup$
– Derek Holt
Dec 6 '13 at 8:49
1
$begingroup$
@DerekHolt The number of non-isomorphic groups of order $n$ is the very first sequence in OEIS! oeis.org/…
$endgroup$
– MartianInvader
Jul 11 '14 at 22:27
|
show 3 more comments
$begingroup$
What do you want to know exactly? This is a very difficult question in general, though it's possible to answer in specific cases using theorems in group theory. For example, when $n$ is prime, there are $ncdot (n-2)!$ possible groups. (This follows from the fact that every group of prime order is cyclic, as well as some counting.)
$endgroup$
– Jim Belk
Dec 6 '13 at 6:51
$begingroup$
I need to know most likely in general case!
$endgroup$
– IremadzeArchil19910311
Dec 6 '13 at 6:52
2
$begingroup$
There is an OEIS entry for the sequence that you're looking for. Starting with $n=1$, the sequence is 1, 2, 3, 16, 30, 480, 840, 22080, 68040, 1088640, 3991680, and so forth.
$endgroup$
– Jim Belk
Dec 6 '13 at 7:00
2
$begingroup$
But usually, different groups mean non-isomorphic groups.
$endgroup$
– Derek Holt
Dec 6 '13 at 8:49
1
$begingroup$
@DerekHolt The number of non-isomorphic groups of order $n$ is the very first sequence in OEIS! oeis.org/…
$endgroup$
– MartianInvader
Jul 11 '14 at 22:27
$begingroup$
What do you want to know exactly? This is a very difficult question in general, though it's possible to answer in specific cases using theorems in group theory. For example, when $n$ is prime, there are $ncdot (n-2)!$ possible groups. (This follows from the fact that every group of prime order is cyclic, as well as some counting.)
$endgroup$
– Jim Belk
Dec 6 '13 at 6:51
$begingroup$
What do you want to know exactly? This is a very difficult question in general, though it's possible to answer in specific cases using theorems in group theory. For example, when $n$ is prime, there are $ncdot (n-2)!$ possible groups. (This follows from the fact that every group of prime order is cyclic, as well as some counting.)
$endgroup$
– Jim Belk
Dec 6 '13 at 6:51
$begingroup$
I need to know most likely in general case!
$endgroup$
– IremadzeArchil19910311
Dec 6 '13 at 6:52
$begingroup$
I need to know most likely in general case!
$endgroup$
– IremadzeArchil19910311
Dec 6 '13 at 6:52
2
2
$begingroup$
There is an OEIS entry for the sequence that you're looking for. Starting with $n=1$, the sequence is 1, 2, 3, 16, 30, 480, 840, 22080, 68040, 1088640, 3991680, and so forth.
$endgroup$
– Jim Belk
Dec 6 '13 at 7:00
$begingroup$
There is an OEIS entry for the sequence that you're looking for. Starting with $n=1$, the sequence is 1, 2, 3, 16, 30, 480, 840, 22080, 68040, 1088640, 3991680, and so forth.
$endgroup$
– Jim Belk
Dec 6 '13 at 7:00
2
2
$begingroup$
But usually, different groups mean non-isomorphic groups.
$endgroup$
– Derek Holt
Dec 6 '13 at 8:49
$begingroup$
But usually, different groups mean non-isomorphic groups.
$endgroup$
– Derek Holt
Dec 6 '13 at 8:49
1
1
$begingroup$
@DerekHolt The number of non-isomorphic groups of order $n$ is the very first sequence in OEIS! oeis.org/…
$endgroup$
– MartianInvader
Jul 11 '14 at 22:27
$begingroup$
@DerekHolt The number of non-isomorphic groups of order $n$ is the very first sequence in OEIS! oeis.org/…
$endgroup$
– MartianInvader
Jul 11 '14 at 22:27
|
show 3 more comments
1 Answer
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active
oldest
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$begingroup$
Interesting but very difficult question; sadly I don't think that we will see a definite answer to this question. I think it is clear, considering the complexity of the problem, that we cannot hope for a simple formula for general $n$. (In case you don't believe this, a problem which is rather easier than this one: there is an explicit expression (due to Rademacher) for the partition number.). The best I can do is quote the following (taken from Suzuki: Group theory, vol. I., Grundlehren d. math. Wiss., chap. I, §7.):
Instead of repeating what he says let me remark that that $Sigma_n$ denotes the symmetric group on $n$ letters. It should be said that this book is rather advanced, and it seems that even though it is about 30 years old you cannot hope for more than some (crude) estimates. (I include the references which he gives there upon request.)
$endgroup$
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1 Answer
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$begingroup$
Interesting but very difficult question; sadly I don't think that we will see a definite answer to this question. I think it is clear, considering the complexity of the problem, that we cannot hope for a simple formula for general $n$. (In case you don't believe this, a problem which is rather easier than this one: there is an explicit expression (due to Rademacher) for the partition number.). The best I can do is quote the following (taken from Suzuki: Group theory, vol. I., Grundlehren d. math. Wiss., chap. I, §7.):
Instead of repeating what he says let me remark that that $Sigma_n$ denotes the symmetric group on $n$ letters. It should be said that this book is rather advanced, and it seems that even though it is about 30 years old you cannot hope for more than some (crude) estimates. (I include the references which he gives there upon request.)
$endgroup$
add a comment |
$begingroup$
Interesting but very difficult question; sadly I don't think that we will see a definite answer to this question. I think it is clear, considering the complexity of the problem, that we cannot hope for a simple formula for general $n$. (In case you don't believe this, a problem which is rather easier than this one: there is an explicit expression (due to Rademacher) for the partition number.). The best I can do is quote the following (taken from Suzuki: Group theory, vol. I., Grundlehren d. math. Wiss., chap. I, §7.):
Instead of repeating what he says let me remark that that $Sigma_n$ denotes the symmetric group on $n$ letters. It should be said that this book is rather advanced, and it seems that even though it is about 30 years old you cannot hope for more than some (crude) estimates. (I include the references which he gives there upon request.)
$endgroup$
add a comment |
$begingroup$
Interesting but very difficult question; sadly I don't think that we will see a definite answer to this question. I think it is clear, considering the complexity of the problem, that we cannot hope for a simple formula for general $n$. (In case you don't believe this, a problem which is rather easier than this one: there is an explicit expression (due to Rademacher) for the partition number.). The best I can do is quote the following (taken from Suzuki: Group theory, vol. I., Grundlehren d. math. Wiss., chap. I, §7.):
Instead of repeating what he says let me remark that that $Sigma_n$ denotes the symmetric group on $n$ letters. It should be said that this book is rather advanced, and it seems that even though it is about 30 years old you cannot hope for more than some (crude) estimates. (I include the references which he gives there upon request.)
$endgroup$
Interesting but very difficult question; sadly I don't think that we will see a definite answer to this question. I think it is clear, considering the complexity of the problem, that we cannot hope for a simple formula for general $n$. (In case you don't believe this, a problem which is rather easier than this one: there is an explicit expression (due to Rademacher) for the partition number.). The best I can do is quote the following (taken from Suzuki: Group theory, vol. I., Grundlehren d. math. Wiss., chap. I, §7.):
Instead of repeating what he says let me remark that that $Sigma_n$ denotes the symmetric group on $n$ letters. It should be said that this book is rather advanced, and it seems that even though it is about 30 years old you cannot hope for more than some (crude) estimates. (I include the references which he gives there upon request.)
edited Jul 14 '14 at 20:19
answered Jul 14 '14 at 19:51
user164074
add a comment |
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$begingroup$
What do you want to know exactly? This is a very difficult question in general, though it's possible to answer in specific cases using theorems in group theory. For example, when $n$ is prime, there are $ncdot (n-2)!$ possible groups. (This follows from the fact that every group of prime order is cyclic, as well as some counting.)
$endgroup$
– Jim Belk
Dec 6 '13 at 6:51
$begingroup$
I need to know most likely in general case!
$endgroup$
– IremadzeArchil19910311
Dec 6 '13 at 6:52
2
$begingroup$
There is an OEIS entry for the sequence that you're looking for. Starting with $n=1$, the sequence is 1, 2, 3, 16, 30, 480, 840, 22080, 68040, 1088640, 3991680, and so forth.
$endgroup$
– Jim Belk
Dec 6 '13 at 7:00
2
$begingroup$
But usually, different groups mean non-isomorphic groups.
$endgroup$
– Derek Holt
Dec 6 '13 at 8:49
1
$begingroup$
@DerekHolt The number of non-isomorphic groups of order $n$ is the very first sequence in OEIS! oeis.org/…
$endgroup$
– MartianInvader
Jul 11 '14 at 22:27