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What is $tau(A_n)$?
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Suppose G is a finite group. Define $tau(G)$ as the minimal number, such that $forall X subset G$ if $|X| > tau(G)$, then $XXX = langle X rangle$.
What is $tau(A_n)$?
Similar problems for some different classes of groups are already answered:
1) $tau(mathbbZ_n) = lceil fracn3 rceil + 1$ (this is a number-theoretic fact proved via arithmetic progressions)
2) Gowers, Nikolov and Pyber proved the fact that $tau(SL_n(mathbbZ_p)) = 2|SL_n(mathbbZ_p)|^1-frac13(n+1)$ (this fact is proved with linear algebra)
However, I have never seen anything like that for $A_n$. It will be interesting to know if there is something...
abstract-algebra group-theory finite-groups permutations additive-combinatorics
asked Jul 19 at 20:54
Yanior Weg
1,0211629
This question has an open bounty worth +100
reputation from Yanior Weg ending ending at 2018-08-22 13:02:32Z">in 6 days.
This question has not received enough attention.
add a comment |Â
up vote
23
down vote
favorite
Suppose G is a finite group. Define $tau(G)$ as the minimal number, such that $forall X subset G$ if $|X| > tau(G)$, then $XXX = langle X rangle$.
What is $tau(A_n)$?
Similar problems for some different classes of groups are already answered:
1) $tau(mathbbZ_n) = lceil fracn3 rceil + 1$ (this is a number-theoretic fact proved via arithmetic progressions)
2) Gowers, Nikolov and Pyber proved the fact that $tau(SL_n(mathbbZ_p)) = 2|SL_n(mathbbZ_p)|^1-frac13(n+1)$ (this fact is proved with linear algebra)
However, I have never seen anything like that for $A_n$. It will be interesting to know if there is something...
abstract-algebra group-theory finite-groups permutations additive-combinatorics
asked Jul 19 at 20:54
Yanior Weg
1,0211629
This question has an open bounty worth +100
reputation from Yanior Weg ending ending at 2018-08-22 13:02:32Z">in 6 days.
This question has not received enough attention.
I don't even understand the question. What do you mean when you say $XXX=langleXrangle$?
– crskhr
Aug 6 at 13:44
@crskhr, $XXX$ stands for the group subset product: $XXX = abc$; $langle X rangle$ stands for group subset closure: $langle X rangle$ is the minimal subgroup that contains $X$.
– Yanior Weg
Aug 6 at 13:52
add a comment |Â
up vote
23
down vote
favorite
up vote
23
down vote
favorite
Suppose G is a finite group. Define $tau(G)$ as the minimal number, such that $forall X subset G$ if $|X| > tau(G)$, then $XXX = langle X rangle$.
What is $tau(A_n)$?
Similar problems for some different classes of groups are already answered:
1) $tau(mathbbZ_n) = lceil fracn3 rceil + 1$ (this is a number-theoretic fact proved via arithmetic progressions)
2) Gowers, Nikolov and Pyber proved the fact that $tau(SL_n(mathbbZ_p)) = 2|SL_n(mathbbZ_p)|^1-frac13(n+1)$ (this fact is proved with linear algebra)
However, I have never seen anything like that for $A_n$. It will be interesting to know if there is something...
abstract-algebra group-theory finite-groups permutations additive-combinatorics
asked Jul 19 at 20:54
Yanior Weg
1,0211629
Suppose G is a finite group. Define $tau(G)$ as the minimal number, such that $forall X subset G$ if $|X| > tau(G)$, then $XXX = langle X rangle$.
What is $tau(A_n)$?
Similar problems for some different classes of groups are already answered:
1) $tau(mathbbZ_n) = lceil fracn3 rceil + 1$ (this is a number-theoretic fact proved via arithmetic progressions)
2) Gowers, Nikolov and Pyber proved the fact that $tau(SL_n(mathbbZ_p)) = 2|SL_n(mathbbZ_p)|^1-frac13(n+1)$ (this fact is proved with linear algebra)
However, I have never seen anything like that for $A_n$. It will be interesting to know if there is something...
abstract-algebra group-theory finite-groups permutations additive-combinatorics
asked Jul 19 at 20:54
Yanior Weg
1,0211629
edited Aug 6 at 13:08
asked Jul 19 at 20:54
Yanior Weg
1,0211629
asked Jul 19 at 20:54
Yanior Weg
1,0211629
asked Jul 19 at 20:54
Yanior Weg
1,0211629
1,0211629
This question has an open bounty worth +100
reputation from Yanior Weg ending ending at 2018-08-22 13:02:32Z">in 6 days.
This question has not received enough attention.
This question has an open bounty worth +100
reputation from Yanior Weg ending ending at 2018-08-22 13:02:32Z">in 6 days.
This question has not received enough attention.
I don't even understand the question. What do you mean when you say $XXX=langleXrangle$?
– crskhr
Aug 6 at 13:44
@crskhr, $XXX$ stands for the group subset product: $XXX = abc$; $langle X rangle$ stands for group subset closure: $langle X rangle$ is the minimal subgroup that contains $X$.
– Yanior Weg
Aug 6 at 13:52
add a comment |Â
I don't even understand the question. What do you mean when you say $XXX=langleXrangle$?
– crskhr
Aug 6 at 13:44
@crskhr, $XXX$ stands for the group subset product: $XXX = abc$; $langle X rangle$ stands for group subset closure: $langle X rangle$ is the minimal subgroup that contains $X$.
– Yanior Weg
Aug 6 at 13:52
I don't even understand the question. What do you mean when you say $XXX=langleXrangle$?
– crskhr
Aug 6 at 13:44
I don't even understand the question. What do you mean when you say $XXX=langleXrangle$?
– crskhr
Aug 6 at 13:44
@crskhr, $XXX$ stands for the group subset product: $XXX = abc$; $langle X rangle$ stands for group subset closure: $langle X rangle$ is the minimal subgroup that contains $X$.
– Yanior Weg
Aug 6 at 13:52
@crskhr, $XXX$ stands for the group subset product: $XXX = abc$; $langle X rangle$ stands for group subset closure: $langle X rangle$ is the minimal subgroup that contains $X$.
– Yanior Weg
Aug 6 at 13:52
add a comment |Â
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I don't even understand the question. What do you mean when you say $XXX=langleXrangle$?
– crskhr
Aug 6 at 13:44
@crskhr, $XXX$ stands for the group subset product: $XXX = abc$; $langle X rangle$ stands for group subset closure: $langle X rangle$ is the minimal subgroup that contains $X$.
– Yanior Weg
Aug 6 at 13:52