A measure of a group's commutativity
$begingroup$
Let $G$ be a finitely presented group on and let $ma(G) = n$ be the minimum of $left| K right|$ ($K subseteq G$) over all group presentations $G cong F{A} / H$ such that $G / {H vee [K, K]}$ is commutative.
If $ma(G) = 0$ then $G$ is abelian and if $G = F{A}$ then $ma(G) = left|Aright|$.
Thus $ma$ measures the the numbers of 'steps' $G$ is away from being commutative.
Is this construction useful? Does it have a name?
group-theory terminology definition abelian-groups group-presentation
edited Nov 29 '18 at 20:14
Shaun
8,888113681
asked Jul 9 '15 at 17:34
ThoralfSkolemThoralfSkolem
1,133615
$endgroup$
add a comment |
$begingroup$
Let $G$ be a finitely presented group on and let $ma(G) = n$ be the minimum of $left| K right|$ ($K subseteq G$) over all group presentations $G cong F{A} / H$ such that $G / {H vee [K, K]}$ is commutative.
If $ma(G) = 0$ then $G$ is abelian and if $G = F{A}$ then $ma(G) = left|Aright|$.
Thus $ma$ measures the the numbers of 'steps' $G$ is away from being commutative.
Is this construction useful? Does it have a name?
group-theory terminology definition abelian-groups group-presentation
edited Nov 29 '18 at 20:14
Shaun
8,888113681
asked Jul 9 '15 at 17:34
ThoralfSkolemThoralfSkolem
1,133615
$endgroup$
$begingroup$
Please excuse my ignorance here. What do you mean by $F{ A}$?
$endgroup$
– Shaun
Nov 29 '18 at 20:12
$begingroup$
Do you mean, say, the free group over the set $A$?
$endgroup$
– Shaun
Nov 29 '18 at 20:15
$begingroup$
Yes, the free group over the set A.
$endgroup$
– ThoralfSkolem
Nov 30 '18 at 22:52
$begingroup$
I thought so, @ThoralfSkolem. When I think about measuring commutativity, by the way, I'm reminded of Lie algebras. It might be worth adding thelie-algebrastag.
$endgroup$
– Shaun
Nov 30 '18 at 23:14
add a comment |
$begingroup$
Let $G$ be a finitely presented group on and let $ma(G) = n$ be the minimum of $left| K right|$ ($K subseteq G$) over all group presentations $G cong F{A} / H$ such that $G / {H vee [K, K]}$ is commutative.
If $ma(G) = 0$ then $G$ is abelian and if $G = F{A}$ then $ma(G) = left|Aright|$.
Thus $ma$ measures the the numbers of 'steps' $G$ is away from being commutative.
Is this construction useful? Does it have a name?
group-theory terminology definition abelian-groups group-presentation
edited Nov 29 '18 at 20:14
Shaun
8,888113681
asked Jul 9 '15 at 17:34
ThoralfSkolemThoralfSkolem
1,133615
$endgroup$
Let $G$ be a finitely presented group on and let $ma(G) = n$ be the minimum of $left| K right|$ ($K subseteq G$) over all group presentations $G cong F{A} / H$ such that $G / {H vee [K, K]}$ is commutative.
If $ma(G) = 0$ then $G$ is abelian and if $G = F{A}$ then $ma(G) = left|Aright|$.
Thus $ma$ measures the the numbers of 'steps' $G$ is away from being commutative.
Is this construction useful? Does it have a name?
group-theory terminology definition abelian-groups group-presentation
group-theory terminology definition abelian-groups group-presentation
edited Nov 29 '18 at 20:14
Shaun
8,888113681
asked Jul 9 '15 at 17:34
ThoralfSkolemThoralfSkolem
1,133615
edited Nov 29 '18 at 20:14
Shaun
8,888113681
asked Jul 9 '15 at 17:34
ThoralfSkolemThoralfSkolem
1,133615
edited Nov 29 '18 at 20:14
Shaun
8,888113681
edited Nov 29 '18 at 20:14
Shaun
8,888113681
edited Nov 29 '18 at 20:14
Shaun
8,888113681
8,888113681
asked Jul 9 '15 at 17:34
ThoralfSkolemThoralfSkolem
1,133615
asked Jul 9 '15 at 17:34
ThoralfSkolemThoralfSkolem
1,133615
asked Jul 9 '15 at 17:34
ThoralfSkolemThoralfSkolem
1,133615
1,133615
$begingroup$
Please excuse my ignorance here. What do you mean by $F{ A}$?
$endgroup$
– Shaun
Nov 29 '18 at 20:12
$begingroup$
Do you mean, say, the free group over the set $A$?
$endgroup$
– Shaun
Nov 29 '18 at 20:15
$begingroup$
Yes, the free group over the set A.
$endgroup$
– ThoralfSkolem
Nov 30 '18 at 22:52
$begingroup$
I thought so, @ThoralfSkolem. When I think about measuring commutativity, by the way, I'm reminded of Lie algebras. It might be worth adding thelie-algebrastag.
$endgroup$
– Shaun
Nov 30 '18 at 23:14
add a comment |
$begingroup$
Please excuse my ignorance here. What do you mean by $F{ A}$?
$endgroup$
– Shaun
Nov 29 '18 at 20:12
$begingroup$
Do you mean, say, the free group over the set $A$?
$endgroup$
– Shaun
Nov 29 '18 at 20:15
$begingroup$
Yes, the free group over the set A.
$endgroup$
– ThoralfSkolem
Nov 30 '18 at 22:52
$begingroup$
I thought so, @ThoralfSkolem. When I think about measuring commutativity, by the way, I'm reminded of Lie algebras. It might be worth adding thelie-algebrastag.
$endgroup$
– Shaun
Nov 30 '18 at 23:14
$begingroup$
Please excuse my ignorance here. What do you mean by $F{ A}$?
$endgroup$
– Shaun
Nov 29 '18 at 20:12
$begingroup$
Please excuse my ignorance here. What do you mean by $F{ A}$?
$endgroup$
– Shaun
Nov 29 '18 at 20:12
$begingroup$
Do you mean, say, the free group over the set $A$?
$endgroup$
– Shaun
Nov 29 '18 at 20:15
$begingroup$
Do you mean, say, the free group over the set $A$?
$endgroup$
– Shaun
Nov 29 '18 at 20:15
$begingroup$
Yes, the free group over the set A.
$endgroup$
– ThoralfSkolem
Nov 30 '18 at 22:52
$begingroup$
Yes, the free group over the set A.
$endgroup$
– ThoralfSkolem
Nov 30 '18 at 22:52
$begingroup$
I thought so, @ThoralfSkolem. When I think about measuring commutativity, by the way, I'm reminded of Lie algebras. It might be worth adding the
lie-algebras tag.$endgroup$
– Shaun
Nov 30 '18 at 23:14
$begingroup$
I thought so, @ThoralfSkolem. When I think about measuring commutativity, by the way, I'm reminded of Lie algebras. It might be worth adding the
lie-algebras tag.$endgroup$
– Shaun
Nov 30 '18 at 23:14
add a comment |
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$begingroup$
Please excuse my ignorance here. What do you mean by $F{ A}$?
$endgroup$
– Shaun
Nov 29 '18 at 20:12
$begingroup$
Do you mean, say, the free group over the set $A$?
$endgroup$
– Shaun
Nov 29 '18 at 20:15
$begingroup$
Yes, the free group over the set A.
$endgroup$
– ThoralfSkolem
Nov 30 '18 at 22:52
$begingroup$
I thought so, @ThoralfSkolem. When I think about measuring commutativity, by the way, I'm reminded of Lie algebras. It might be worth adding the
lie-algebrastag.$endgroup$
– Shaun
Nov 30 '18 at 23:14