Can Erdős-Turán $frac{5}{8}$ theorem be generalised that way?
$begingroup$
Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $forall a_1, … , a_n in G$ $w(a_1, … , a_n) = e$. Is it true, that for any group word $w$ there exists a positive real number $epsilon (w) > 0$, such that any finite group $G$ is in $mathfrak{U_w}$ iff $$frac{lvert{(a_1, … , a_n) in G^n : w(a_1, … , a_n) = e}rvert}{{|G|}^n} > 1 - epsilon(w)?$$
How did this question arise? There is a widely known theorem proved by P. Erdős and P. Turán that states:
A finite group $G$ is abelian iff $$frac{|{(a, b) in G^2 : [a, b] = e}|}{{|G|}^2} > frac{5}{8}.$$
This theorem can be rephrased using aforementioned terminology as $epsilon([a, b]) = frac{3}{8}$.
There also is a generalisation of this theorem, stating that a finite group $G$ is nilpotent of class $n$ iff $$frac{|{(a_0, a_1, … , a_n) in G^{n + 1} : [ … [[a_0, a_1], a_2]… a_n] = e}|}{{|G|}^{n + 1}} > 1 - frac{3}{2^{n + 2}},$$ thus making $epsilon([ … [[a_0, a_1], a_2]… a_n]) = frac{3}{2^{n + 2}}$.
However, I have never seen similar statements about other one-word varieties being proved or disproved, despite such question seeming quite natural . . .
Actually, I doubt that the conjecture in the main part of question is true. However, I failed to find any counterexamples myself.
combinatorics group-theory finite-groups conjectures group-varieties
edited Jan 22 at 17:12
user1729
16.9k64085
asked Jan 11 at 20:51
Yanior WegYanior Weg
1,30311136
$endgroup$
add a comment |
$begingroup$
Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $forall a_1, … , a_n in G$ $w(a_1, … , a_n) = e$. Is it true, that for any group word $w$ there exists a positive real number $epsilon (w) > 0$, such that any finite group $G$ is in $mathfrak{U_w}$ iff $$frac{lvert{(a_1, … , a_n) in G^n : w(a_1, … , a_n) = e}rvert}{{|G|}^n} > 1 - epsilon(w)?$$
How did this question arise? There is a widely known theorem proved by P. Erdős and P. Turán that states:
A finite group $G$ is abelian iff $$frac{|{(a, b) in G^2 : [a, b] = e}|}{{|G|}^2} > frac{5}{8}.$$
This theorem can be rephrased using aforementioned terminology as $epsilon([a, b]) = frac{3}{8}$.
There also is a generalisation of this theorem, stating that a finite group $G$ is nilpotent of class $n$ iff $$frac{|{(a_0, a_1, … , a_n) in G^{n + 1} : [ … [[a_0, a_1], a_2]… a_n] = e}|}{{|G|}^{n + 1}} > 1 - frac{3}{2^{n + 2}},$$ thus making $epsilon([ … [[a_0, a_1], a_2]… a_n]) = frac{3}{2^{n + 2}}$.
However, I have never seen similar statements about other one-word varieties being proved or disproved, despite such question seeming quite natural . . .
Actually, I doubt that the conjecture in the main part of question is true. However, I failed to find any counterexamples myself.
combinatorics group-theory finite-groups conjectures group-varieties
edited Jan 22 at 17:12
user1729
16.9k64085
asked Jan 11 at 20:51
Yanior WegYanior Weg
1,30311136
$endgroup$
1
$begingroup$
I one wrote out an answer which is about generalising the Erdos-Turan result to infinite groups: math.stackexchange.com/a/2809964/10513 You might find it interesting/relevant.
$endgroup$
– user1729
Jan 18 at 11:54
3
$begingroup$
Not mentioned in a 2015 survey Farrokhi, D. G. (2015). ON THE PROBABILITY THAT A GROUP SATISFIES A LAW: A SURVEY (Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics), muroran-it.ac.jp/mathsci/danwakai/past/articles/201404-201503/…. Mentioned as open in a note by John D. Dixon, "Probabilistic Group Theory", people.math.carleton.ca/~jdixon/Prgrpth.pdf
$endgroup$
– Dap
Jan 18 at 18:00
$begingroup$
Is the $n=1$ case obviously true? Or is even that case difficult?
$endgroup$
– Mees de Vries
2 days ago
add a comment |
$begingroup$
Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $forall a_1, … , a_n in G$ $w(a_1, … , a_n) = e$. Is it true, that for any group word $w$ there exists a positive real number $epsilon (w) > 0$, such that any finite group $G$ is in $mathfrak{U_w}$ iff $$frac{lvert{(a_1, … , a_n) in G^n : w(a_1, … , a_n) = e}rvert}{{|G|}^n} > 1 - epsilon(w)?$$
How did this question arise? There is a widely known theorem proved by P. Erdős and P. Turán that states:
A finite group $G$ is abelian iff $$frac{|{(a, b) in G^2 : [a, b] = e}|}{{|G|}^2} > frac{5}{8}.$$
This theorem can be rephrased using aforementioned terminology as $epsilon([a, b]) = frac{3}{8}$.
There also is a generalisation of this theorem, stating that a finite group $G$ is nilpotent of class $n$ iff $$frac{|{(a_0, a_1, … , a_n) in G^{n + 1} : [ … [[a_0, a_1], a_2]… a_n] = e}|}{{|G|}^{n + 1}} > 1 - frac{3}{2^{n + 2}},$$ thus making $epsilon([ … [[a_0, a_1], a_2]… a_n]) = frac{3}{2^{n + 2}}$.
However, I have never seen similar statements about other one-word varieties being proved or disproved, despite such question seeming quite natural . . .
Actually, I doubt that the conjecture in the main part of question is true. However, I failed to find any counterexamples myself.
combinatorics group-theory finite-groups conjectures group-varieties
edited Jan 22 at 17:12
user1729
16.9k64085
asked Jan 11 at 20:51
Yanior WegYanior Weg
1,30311136
$endgroup$
Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $forall a_1, … , a_n in G$ $w(a_1, … , a_n) = e$. Is it true, that for any group word $w$ there exists a positive real number $epsilon (w) > 0$, such that any finite group $G$ is in $mathfrak{U_w}$ iff $$frac{lvert{(a_1, … , a_n) in G^n : w(a_1, … , a_n) = e}rvert}{{|G|}^n} > 1 - epsilon(w)?$$
How did this question arise? There is a widely known theorem proved by P. Erdős and P. Turán that states:
A finite group $G$ is abelian iff $$frac{|{(a, b) in G^2 : [a, b] = e}|}{{|G|}^2} > frac{5}{8}.$$
This theorem can be rephrased using aforementioned terminology as $epsilon([a, b]) = frac{3}{8}$.
There also is a generalisation of this theorem, stating that a finite group $G$ is nilpotent of class $n$ iff $$frac{|{(a_0, a_1, … , a_n) in G^{n + 1} : [ … [[a_0, a_1], a_2]… a_n] = e}|}{{|G|}^{n + 1}} > 1 - frac{3}{2^{n + 2}},$$ thus making $epsilon([ … [[a_0, a_1], a_2]… a_n]) = frac{3}{2^{n + 2}}$.
However, I have never seen similar statements about other one-word varieties being proved or disproved, despite such question seeming quite natural . . .
Actually, I doubt that the conjecture in the main part of question is true. However, I failed to find any counterexamples myself.
combinatorics group-theory finite-groups conjectures group-varieties
combinatorics group-theory finite-groups conjectures group-varieties
edited Jan 22 at 17:12
user1729
16.9k64085
asked Jan 11 at 20:51
Yanior WegYanior Weg
1,30311136
edited Jan 22 at 17:12
user1729
16.9k64085
asked Jan 11 at 20:51
Yanior WegYanior Weg
1,30311136
edited Jan 22 at 17:12
user1729
16.9k64085
edited Jan 22 at 17:12
user1729
16.9k64085
edited Jan 22 at 17:12
user1729
16.9k64085
16.9k64085
asked Jan 11 at 20:51
Yanior WegYanior Weg
1,30311136
asked Jan 11 at 20:51
Yanior WegYanior Weg
1,30311136
asked Jan 11 at 20:51
Yanior WegYanior Weg
1,30311136
1,30311136
1
$begingroup$
I one wrote out an answer which is about generalising the Erdos-Turan result to infinite groups: math.stackexchange.com/a/2809964/10513 You might find it interesting/relevant.
$endgroup$
– user1729
Jan 18 at 11:54
3
$begingroup$
Not mentioned in a 2015 survey Farrokhi, D. G. (2015). ON THE PROBABILITY THAT A GROUP SATISFIES A LAW: A SURVEY (Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics), muroran-it.ac.jp/mathsci/danwakai/past/articles/201404-201503/…. Mentioned as open in a note by John D. Dixon, "Probabilistic Group Theory", people.math.carleton.ca/~jdixon/Prgrpth.pdf
$endgroup$
– Dap
Jan 18 at 18:00
$begingroup$
Is the $n=1$ case obviously true? Or is even that case difficult?
$endgroup$
– Mees de Vries
2 days ago
add a comment |
1
$begingroup$
I one wrote out an answer which is about generalising the Erdos-Turan result to infinite groups: math.stackexchange.com/a/2809964/10513 You might find it interesting/relevant.
$endgroup$
– user1729
Jan 18 at 11:54
3
$begingroup$
Not mentioned in a 2015 survey Farrokhi, D. G. (2015). ON THE PROBABILITY THAT A GROUP SATISFIES A LAW: A SURVEY (Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics), muroran-it.ac.jp/mathsci/danwakai/past/articles/201404-201503/…. Mentioned as open in a note by John D. Dixon, "Probabilistic Group Theory", people.math.carleton.ca/~jdixon/Prgrpth.pdf
$endgroup$
– Dap
Jan 18 at 18:00
$begingroup$
Is the $n=1$ case obviously true? Or is even that case difficult?
$endgroup$
– Mees de Vries
2 days ago
1
1
$begingroup$
I one wrote out an answer which is about generalising the Erdos-Turan result to infinite groups: math.stackexchange.com/a/2809964/10513 You might find it interesting/relevant.
$endgroup$
– user1729
Jan 18 at 11:54
$begingroup$
I one wrote out an answer which is about generalising the Erdos-Turan result to infinite groups: math.stackexchange.com/a/2809964/10513 You might find it interesting/relevant.
$endgroup$
– user1729
Jan 18 at 11:54
3
3
$begingroup$
Not mentioned in a 2015 survey Farrokhi, D. G. (2015). ON THE PROBABILITY THAT A GROUP SATISFIES A LAW: A SURVEY (Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics), muroran-it.ac.jp/mathsci/danwakai/past/articles/201404-201503/…. Mentioned as open in a note by John D. Dixon, "Probabilistic Group Theory", people.math.carleton.ca/~jdixon/Prgrpth.pdf
$endgroup$
– Dap
Jan 18 at 18:00
$begingroup$
Not mentioned in a 2015 survey Farrokhi, D. G. (2015). ON THE PROBABILITY THAT A GROUP SATISFIES A LAW: A SURVEY (Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics), muroran-it.ac.jp/mathsci/danwakai/past/articles/201404-201503/…. Mentioned as open in a note by John D. Dixon, "Probabilistic Group Theory", people.math.carleton.ca/~jdixon/Prgrpth.pdf
$endgroup$
– Dap
Jan 18 at 18:00
$begingroup$
Is the $n=1$ case obviously true? Or is even that case difficult?
$endgroup$
– Mees de Vries
2 days ago
$begingroup$
Is the $n=1$ case obviously true? Or is even that case difficult?
$endgroup$
– Mees de Vries
2 days ago
add a comment |
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$begingroup$
I one wrote out an answer which is about generalising the Erdos-Turan result to infinite groups: math.stackexchange.com/a/2809964/10513 You might find it interesting/relevant.
$endgroup$
– user1729
Jan 18 at 11:54
3
$begingroup$
Not mentioned in a 2015 survey Farrokhi, D. G. (2015). ON THE PROBABILITY THAT A GROUP SATISFIES A LAW: A SURVEY (Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics), muroran-it.ac.jp/mathsci/danwakai/past/articles/201404-201503/…. Mentioned as open in a note by John D. Dixon, "Probabilistic Group Theory", people.math.carleton.ca/~jdixon/Prgrpth.pdf
$endgroup$
– Dap
Jan 18 at 18:00
$begingroup$
Is the $n=1$ case obviously true? Or is even that case difficult?
$endgroup$
– Mees de Vries
2 days ago