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Probability and Combinatorial Group Theory.
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If this is too broad or is otherwise a poor question, I apologise.
I learned recently that the probability that two integers generate the additive group of integers is $frac6pi^2$.
What other results are there like this?
I'm looking for any results of probability applied to group theory, preferably combinatorial group theory, in manner such as the one above.
probability group-theory big-list combinatorial-group-theory
asked Jan 9 at 23:49
Shaun
7,30092870
This question has an open bounty worth +100
reputation from Yanior Weg ending ending at 2018-08-12 13:20:56Z">in 7 days.
This question has not received enough attention.
 |Â
show 4 more comments
up vote
3
down vote
favorite
If this is too broad or is otherwise a poor question, I apologise.
I learned recently that the probability that two integers generate the additive group of integers is $frac6pi^2$.
What other results are there like this?
I'm looking for any results of probability applied to group theory, preferably combinatorial group theory, in manner such as the one above.
probability group-theory big-list combinatorial-group-theory
asked Jan 9 at 23:49
Shaun
7,30092870
This question has an open bounty worth +100
reputation from Yanior Weg ending ending at 2018-08-12 13:20:56Z">in 7 days.
This question has not received enough attention.
Related.
– Shaun
Jan 9 at 23:57
1
The buzzwords you want are “Probabalistic†and “Asymptotoc†group theory.
– ml0105
Jan 10 at 0:15
Related (PDF): in it, it is claim that the probability that any two elements of $S_n$ generate $S_n$ approaches $1$ as $ntoinfty$.
– Shaun
Jan 10 at 0:55
Thank you, @ml0105; Google comes up with plenty of results using "probabilistic group theory" alone. I swear, I did some searching before posting. It just never occurred to me to use those words.
– Shaun
Jan 10 at 1:00
1
Related.
– Shaun
Jun 26 at 13:37
 |Â
show 4 more comments
up vote
3
down vote
favorite
up vote
3
down vote
favorite
If this is too broad or is otherwise a poor question, I apologise.
I learned recently that the probability that two integers generate the additive group of integers is $frac6pi^2$.
What other results are there like this?
I'm looking for any results of probability applied to group theory, preferably combinatorial group theory, in manner such as the one above.
probability group-theory big-list combinatorial-group-theory
asked Jan 9 at 23:49
Shaun
7,30092870
If this is too broad or is otherwise a poor question, I apologise.
I learned recently that the probability that two integers generate the additive group of integers is $frac6pi^2$.
What other results are there like this?
I'm looking for any results of probability applied to group theory, preferably combinatorial group theory, in manner such as the one above.
probability group-theory big-list combinatorial-group-theory
asked Jan 9 at 23:49
Shaun
7,30092870
asked Jan 9 at 23:49
Shaun
7,30092870
asked Jan 9 at 23:49
Shaun
7,30092870
asked Jan 9 at 23:49
Shaun
7,30092870
7,30092870
This question has an open bounty worth +100
reputation from Yanior Weg ending ending at 2018-08-12 13:20:56Z">in 7 days.
This question has not received enough attention.
This question has an open bounty worth +100
reputation from Yanior Weg ending ending at 2018-08-12 13:20:56Z">in 7 days.
This question has not received enough attention.
Related.
– Shaun
Jan 9 at 23:57
1
The buzzwords you want are “Probabalistic†and “Asymptotoc†group theory.
– ml0105
Jan 10 at 0:15
Related (PDF): in it, it is claim that the probability that any two elements of $S_n$ generate $S_n$ approaches $1$ as $ntoinfty$.
– Shaun
Jan 10 at 0:55
Thank you, @ml0105; Google comes up with plenty of results using "probabilistic group theory" alone. I swear, I did some searching before posting. It just never occurred to me to use those words.
– Shaun
Jan 10 at 1:00
1
Related.
– Shaun
Jun 26 at 13:37
 |Â
show 4 more comments
Related.
– Shaun
Jan 9 at 23:57
1
The buzzwords you want are “Probabalistic†and “Asymptotoc†group theory.
– ml0105
Jan 10 at 0:15
Related (PDF): in it, it is claim that the probability that any two elements of $S_n$ generate $S_n$ approaches $1$ as $ntoinfty$.
– Shaun
Jan 10 at 0:55
Thank you, @ml0105; Google comes up with plenty of results using "probabilistic group theory" alone. I swear, I did some searching before posting. It just never occurred to me to use those words.
– Shaun
Jan 10 at 1:00
1
Related.
– Shaun
Jun 26 at 13:37
Related.
– Shaun
Jan 9 at 23:57
Related.
– Shaun
Jan 9 at 23:57
1
1
The buzzwords you want are “Probabalistic†and “Asymptotoc†group theory.
– ml0105
Jan 10 at 0:15
The buzzwords you want are “Probabalistic†and “Asymptotoc†group theory.
– ml0105
Jan 10 at 0:15
Related (PDF): in it, it is claim that the probability that any two elements of $S_n$ generate $S_n$ approaches $1$ as $ntoinfty$.
– Shaun
Jan 10 at 0:55
Related (PDF): in it, it is claim that the probability that any two elements of $S_n$ generate $S_n$ approaches $1$ as $ntoinfty$.
– Shaun
Jan 10 at 0:55
Thank you, @ml0105; Google comes up with plenty of results using "probabilistic group theory" alone. I swear, I did some searching before posting. It just never occurred to me to use those words.
– Shaun
Jan 10 at 1:00
Thank you, @ml0105; Google comes up with plenty of results using "probabilistic group theory" alone. I swear, I did some searching before posting. It just never occurred to me to use those words.
– Shaun
Jan 10 at 1:00
1
1
Related.
– Shaun
Jun 26 at 13:37
Related.
– Shaun
Jun 26 at 13:37
 |Â
show 4 more comments
2 Answers
2
active
oldest
votes
up vote
1
down vote
Here are quite similar facts:
The probability, that a random pair of elements generates $S_n$ is $1 - frac1n - frac1n^2 - frac4n^3 - frac23n^4 - frac172n^5 - frac1542n^6 - O(frac1n^7)$.
The probability, that a random triple of elements generates $S_n$ is $1 - frac1n^2 - frac3n^4 - frac6n^5 - O(frac1n^6)$.
The probability, that a random element generate $mathbbZ_n$ is $fracphi(n)n$, where $phi$ is Euler's totient function.
The probability, that $n$ random elements generates $mathbbZ^n-1$ is $prod_j=2^n zeta(j)^-1.$, where $zeta$ is Riemann zeta function.
The probability, that $m$ random elements generate $mathbbZ_p^n$, where $p$ is prime, is $1 - frac1p^m$.
answered Jul 18 at 17:50
Yanior Weg
1,0421426
Another one: Dixon's conjecture—the probability, that two random elements generate a finite simple group $G$ tends to $1$ as $lvert G rvert to infty$.
– Orat
1 hour ago
add a comment |Â
up vote
0
down vote
For a finite group you can compute the probability that any two random elements will commute by looking at certain conjugacy classes.
answered 4 hours ago
Jimmy Mixco
463
1
That sounds interesting... Would you mind to give us a bit more detailed information about this fact?
– Yanior Weg
4 hours 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
1
down vote
Here are quite similar facts:
The probability, that a random pair of elements generates $S_n$ is $1 - frac1n - frac1n^2 - frac4n^3 - frac23n^4 - frac172n^5 - frac1542n^6 - O(frac1n^7)$.
The probability, that a random triple of elements generates $S_n$ is $1 - frac1n^2 - frac3n^4 - frac6n^5 - O(frac1n^6)$.
The probability, that a random element generate $mathbbZ_n$ is $fracphi(n)n$, where $phi$ is Euler's totient function.
The probability, that $n$ random elements generates $mathbbZ^n-1$ is $prod_j=2^n zeta(j)^-1.$, where $zeta$ is Riemann zeta function.
The probability, that $m$ random elements generate $mathbbZ_p^n$, where $p$ is prime, is $1 - frac1p^m$.
answered Jul 18 at 17:50
Yanior Weg
1,0421426
Another one: Dixon's conjecture—the probability, that two random elements generate a finite simple group $G$ tends to $1$ as $lvert G rvert to infty$.
– Orat
1 hour ago
add a comment |Â
up vote
1
down vote
Here are quite similar facts:
The probability, that a random pair of elements generates $S_n$ is $1 - frac1n - frac1n^2 - frac4n^3 - frac23n^4 - frac172n^5 - frac1542n^6 - O(frac1n^7)$.
The probability, that a random triple of elements generates $S_n$ is $1 - frac1n^2 - frac3n^4 - frac6n^5 - O(frac1n^6)$.
The probability, that a random element generate $mathbbZ_n$ is $fracphi(n)n$, where $phi$ is Euler's totient function.
The probability, that $n$ random elements generates $mathbbZ^n-1$ is $prod_j=2^n zeta(j)^-1.$, where $zeta$ is Riemann zeta function.
The probability, that $m$ random elements generate $mathbbZ_p^n$, where $p$ is prime, is $1 - frac1p^m$.
answered Jul 18 at 17:50
Yanior Weg
1,0421426
Another one: Dixon's conjecture—the probability, that two random elements generate a finite simple group $G$ tends to $1$ as $lvert G rvert to infty$.
– Orat
1 hour ago
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Here are quite similar facts:
The probability, that a random pair of elements generates $S_n$ is $1 - frac1n - frac1n^2 - frac4n^3 - frac23n^4 - frac172n^5 - frac1542n^6 - O(frac1n^7)$.
The probability, that a random triple of elements generates $S_n$ is $1 - frac1n^2 - frac3n^4 - frac6n^5 - O(frac1n^6)$.
The probability, that a random element generate $mathbbZ_n$ is $fracphi(n)n$, where $phi$ is Euler's totient function.
The probability, that $n$ random elements generates $mathbbZ^n-1$ is $prod_j=2^n zeta(j)^-1.$, where $zeta$ is Riemann zeta function.
The probability, that $m$ random elements generate $mathbbZ_p^n$, where $p$ is prime, is $1 - frac1p^m$.
answered Jul 18 at 17:50
Yanior Weg
1,0421426
Here are quite similar facts:
The probability, that a random pair of elements generates $S_n$ is $1 - frac1n - frac1n^2 - frac4n^3 - frac23n^4 - frac172n^5 - frac1542n^6 - O(frac1n^7)$.
The probability, that a random triple of elements generates $S_n$ is $1 - frac1n^2 - frac3n^4 - frac6n^5 - O(frac1n^6)$.
The probability, that a random element generate $mathbbZ_n$ is $fracphi(n)n$, where $phi$ is Euler's totient function.
The probability, that $n$ random elements generates $mathbbZ^n-1$ is $prod_j=2^n zeta(j)^-1.$, where $zeta$ is Riemann zeta function.
The probability, that $m$ random elements generate $mathbbZ_p^n$, where $p$ is prime, is $1 - frac1p^m$.
answered Jul 18 at 17:50
Yanior Weg
1,0421426
edited 2 hours ago
answered Jul 18 at 17:50
Yanior Weg
1,0421426
answered Jul 18 at 17:50
Yanior Weg
1,0421426
answered Jul 18 at 17:50
Yanior Weg
1,0421426
1,0421426
Another one: Dixon's conjecture—the probability, that two random elements generate a finite simple group $G$ tends to $1$ as $lvert G rvert to infty$.
– Orat
1 hour ago
add a comment |Â
Another one: Dixon's conjecture—the probability, that two random elements generate a finite simple group $G$ tends to $1$ as $lvert G rvert to infty$.
– Orat
1 hour ago
Another one: Dixon's conjecture—the probability, that two random elements generate a finite simple group $G$ tends to $1$ as $lvert G rvert to infty$.
– Orat
1 hour ago
Another one: Dixon's conjecture—the probability, that two random elements generate a finite simple group $G$ tends to $1$ as $lvert G rvert to infty$.
– Orat
1 hour ago
add a comment |Â
up vote
0
down vote
For a finite group you can compute the probability that any two random elements will commute by looking at certain conjugacy classes.
answered 4 hours ago
Jimmy Mixco
463
1
That sounds interesting... Would you mind to give us a bit more detailed information about this fact?
– Yanior Weg
4 hours ago
add a comment |Â
up vote
0
down vote
For a finite group you can compute the probability that any two random elements will commute by looking at certain conjugacy classes.
answered 4 hours ago
Jimmy Mixco
463
1
That sounds interesting... Would you mind to give us a bit more detailed information about this fact?
– Yanior Weg
4 hours ago
add a comment |Â
up vote
0
down vote
up vote
0
down vote
For a finite group you can compute the probability that any two random elements will commute by looking at certain conjugacy classes.
answered 4 hours ago
Jimmy Mixco
463
For a finite group you can compute the probability that any two random elements will commute by looking at certain conjugacy classes.
answered 4 hours ago
Jimmy Mixco
463
answered 4 hours ago
Jimmy Mixco
463
answered 4 hours ago
Jimmy Mixco
463
answered 4 hours ago
Jimmy Mixco
463
463
1
That sounds interesting... Would you mind to give us a bit more detailed information about this fact?
– Yanior Weg
4 hours ago
add a comment |Â
1
That sounds interesting... Would you mind to give us a bit more detailed information about this fact?
– Yanior Weg
4 hours ago
1
1
That sounds interesting... Would you mind to give us a bit more detailed information about this fact?
– Yanior Weg
4 hours ago
That sounds interesting... Would you mind to give us a bit more detailed information about this fact?
– Yanior Weg
4 hours ago
add a comment |Â
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Related.
– Shaun
Jan 9 at 23:57
1
The buzzwords you want are “Probabalistic†and “Asymptotoc†group theory.
– ml0105
Jan 10 at 0:15
Related (PDF): in it, it is claim that the probability that any two elements of $S_n$ generate $S_n$ approaches $1$ as $ntoinfty$.
– Shaun
Jan 10 at 0:55
Thank you, @ml0105; Google comes up with plenty of results using "probabilistic group theory" alone. I swear, I did some searching before posting. It just never occurred to me to use those words.
– Shaun
Jan 10 at 1:00
1
Related.
– Shaun
Jun 26 at 13:37