Quotient group of the free group on the alphabet of a string.
$begingroup$
For instance, let $Sigma = {a,b}$, and consider $F = F(Sigma)$, the free group.
Let $s$ be a string, say $s = ab$ and consider the subgroup $H leqslant F$, $H = langle t: t nleqslant srangle$ where $leqslant$ on strings indicates "is a substring of". Then is $H$ normal?
Let $h in H$, ie $h nleqslant s$, and let $f in F$. Then $fhf^{-1}$ is certainly in $H$ as you can't form a substring via concatenation if it involves any non substring, so $fhf^{-1} in {t nleqslant s}$ is in the generating set of $H$ so is in $H$. Therefore $F/H$ is a group.
What does $F/H$ look like!? It must be ${epsilon, a,b, ab} approx Bbb{Z}_4$ right?
Does this group have a name in literature?
abstract-algebra group-theory formal-languages free-groups
$endgroup$
add a comment |
$begingroup$
For instance, let $Sigma = {a,b}$, and consider $F = F(Sigma)$, the free group.
Let $s$ be a string, say $s = ab$ and consider the subgroup $H leqslant F$, $H = langle t: t nleqslant srangle$ where $leqslant$ on strings indicates "is a substring of". Then is $H$ normal?
Let $h in H$, ie $h nleqslant s$, and let $f in F$. Then $fhf^{-1}$ is certainly in $H$ as you can't form a substring via concatenation if it involves any non substring, so $fhf^{-1} in {t nleqslant s}$ is in the generating set of $H$ so is in $H$. Therefore $F/H$ is a group.
What does $F/H$ look like!? It must be ${epsilon, a,b, ab} approx Bbb{Z}_4$ right?
Does this group have a name in literature?
abstract-algebra group-theory formal-languages free-groups
$endgroup$
add a comment |
$begingroup$
For instance, let $Sigma = {a,b}$, and consider $F = F(Sigma)$, the free group.
Let $s$ be a string, say $s = ab$ and consider the subgroup $H leqslant F$, $H = langle t: t nleqslant srangle$ where $leqslant$ on strings indicates "is a substring of". Then is $H$ normal?
Let $h in H$, ie $h nleqslant s$, and let $f in F$. Then $fhf^{-1}$ is certainly in $H$ as you can't form a substring via concatenation if it involves any non substring, so $fhf^{-1} in {t nleqslant s}$ is in the generating set of $H$ so is in $H$. Therefore $F/H$ is a group.
What does $F/H$ look like!? It must be ${epsilon, a,b, ab} approx Bbb{Z}_4$ right?
Does this group have a name in literature?
abstract-algebra group-theory formal-languages free-groups
$endgroup$
For instance, let $Sigma = {a,b}$, and consider $F = F(Sigma)$, the free group.
Let $s$ be a string, say $s = ab$ and consider the subgroup $H leqslant F$, $H = langle t: t nleqslant srangle$ where $leqslant$ on strings indicates "is a substring of". Then is $H$ normal?
Let $h in H$, ie $h nleqslant s$, and let $f in F$. Then $fhf^{-1}$ is certainly in $H$ as you can't form a substring via concatenation if it involves any non substring, so $fhf^{-1} in {t nleqslant s}$ is in the generating set of $H$ so is in $H$. Therefore $F/H$ is a group.
What does $F/H$ look like!? It must be ${epsilon, a,b, ab} approx Bbb{Z}_4$ right?
Does this group have a name in literature?
abstract-algebra group-theory formal-languages free-groups
abstract-algebra group-theory formal-languages free-groups
edited Dec 9 '18 at 4:48
Eric Wofsey
184k13212338
184k13212338
asked Mar 18 '17 at 19:12
Hermit with AdjointHermit with Adjoint
9,12052458
9,12052458
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Your subgroup $H$ is all of $F$. For instance, if $s=ab$, then $aba$ and $abab$ are both in $H$, and hence so is $b=(aba)^{-1}(abab)$. Similarly, $ain H$ as well. You can easily construct similar examples for any value of $s$.
$endgroup$
add a comment |
$begingroup$
If you pick $s = ab$ as your string, $h = b^{-1}ab$, as your not substring and $f=b$ as your other string, then
$$fhf^{-1} = bb^{-1}abb^{-1} = a leq ab = s.$$
$endgroup$
$begingroup$
Yeah, I saw that happening. I wonder if it's ever normal.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:11
$begingroup$
@FruitfulApproach Now that I think about it, subgroups share the parent's identity, so "", the empty string should be in $H$, but I think the empty string should also be a substring of every string... If so, then H isn't even a subgroup, because the identity isn't in it. :/ The topic seems interesting enough, though!
$endgroup$
– Andrew
Mar 18 '17 at 22:19
$begingroup$
It's called you "artificialy add it in", as it's trivial enough to do. Actually, $epsilon in H$ automatically, by definition of generated subgroup.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:20
$begingroup$
I'm making an answer. I think $s = ababab$ might shed some light
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:21
$begingroup$
Eh, I somehow missed the $ langle $, and its pair, don't ask me how ... And sure thing. :)
$endgroup$
– Andrew
Mar 18 '17 at 22:25
|
show 1 more comment
Your Answer
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Your subgroup $H$ is all of $F$. For instance, if $s=ab$, then $aba$ and $abab$ are both in $H$, and hence so is $b=(aba)^{-1}(abab)$. Similarly, $ain H$ as well. You can easily construct similar examples for any value of $s$.
$endgroup$
add a comment |
$begingroup$
Your subgroup $H$ is all of $F$. For instance, if $s=ab$, then $aba$ and $abab$ are both in $H$, and hence so is $b=(aba)^{-1}(abab)$. Similarly, $ain H$ as well. You can easily construct similar examples for any value of $s$.
$endgroup$
add a comment |
$begingroup$
Your subgroup $H$ is all of $F$. For instance, if $s=ab$, then $aba$ and $abab$ are both in $H$, and hence so is $b=(aba)^{-1}(abab)$. Similarly, $ain H$ as well. You can easily construct similar examples for any value of $s$.
$endgroup$
Your subgroup $H$ is all of $F$. For instance, if $s=ab$, then $aba$ and $abab$ are both in $H$, and hence so is $b=(aba)^{-1}(abab)$. Similarly, $ain H$ as well. You can easily construct similar examples for any value of $s$.
answered Mar 19 '17 at 1:07
Eric WofseyEric Wofsey
184k13212338
184k13212338
add a comment |
add a comment |
$begingroup$
If you pick $s = ab$ as your string, $h = b^{-1}ab$, as your not substring and $f=b$ as your other string, then
$$fhf^{-1} = bb^{-1}abb^{-1} = a leq ab = s.$$
$endgroup$
$begingroup$
Yeah, I saw that happening. I wonder if it's ever normal.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:11
$begingroup$
@FruitfulApproach Now that I think about it, subgroups share the parent's identity, so "", the empty string should be in $H$, but I think the empty string should also be a substring of every string... If so, then H isn't even a subgroup, because the identity isn't in it. :/ The topic seems interesting enough, though!
$endgroup$
– Andrew
Mar 18 '17 at 22:19
$begingroup$
It's called you "artificialy add it in", as it's trivial enough to do. Actually, $epsilon in H$ automatically, by definition of generated subgroup.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:20
$begingroup$
I'm making an answer. I think $s = ababab$ might shed some light
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:21
$begingroup$
Eh, I somehow missed the $ langle $, and its pair, don't ask me how ... And sure thing. :)
$endgroup$
– Andrew
Mar 18 '17 at 22:25
|
show 1 more comment
$begingroup$
If you pick $s = ab$ as your string, $h = b^{-1}ab$, as your not substring and $f=b$ as your other string, then
$$fhf^{-1} = bb^{-1}abb^{-1} = a leq ab = s.$$
$endgroup$
$begingroup$
Yeah, I saw that happening. I wonder if it's ever normal.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:11
$begingroup$
@FruitfulApproach Now that I think about it, subgroups share the parent's identity, so "", the empty string should be in $H$, but I think the empty string should also be a substring of every string... If so, then H isn't even a subgroup, because the identity isn't in it. :/ The topic seems interesting enough, though!
$endgroup$
– Andrew
Mar 18 '17 at 22:19
$begingroup$
It's called you "artificialy add it in", as it's trivial enough to do. Actually, $epsilon in H$ automatically, by definition of generated subgroup.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:20
$begingroup$
I'm making an answer. I think $s = ababab$ might shed some light
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:21
$begingroup$
Eh, I somehow missed the $ langle $, and its pair, don't ask me how ... And sure thing. :)
$endgroup$
– Andrew
Mar 18 '17 at 22:25
|
show 1 more comment
$begingroup$
If you pick $s = ab$ as your string, $h = b^{-1}ab$, as your not substring and $f=b$ as your other string, then
$$fhf^{-1} = bb^{-1}abb^{-1} = a leq ab = s.$$
$endgroup$
If you pick $s = ab$ as your string, $h = b^{-1}ab$, as your not substring and $f=b$ as your other string, then
$$fhf^{-1} = bb^{-1}abb^{-1} = a leq ab = s.$$
answered Mar 18 '17 at 19:51
AndrewAndrew
2,45021324
2,45021324
$begingroup$
Yeah, I saw that happening. I wonder if it's ever normal.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:11
$begingroup$
@FruitfulApproach Now that I think about it, subgroups share the parent's identity, so "", the empty string should be in $H$, but I think the empty string should also be a substring of every string... If so, then H isn't even a subgroup, because the identity isn't in it. :/ The topic seems interesting enough, though!
$endgroup$
– Andrew
Mar 18 '17 at 22:19
$begingroup$
It's called you "artificialy add it in", as it's trivial enough to do. Actually, $epsilon in H$ automatically, by definition of generated subgroup.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:20
$begingroup$
I'm making an answer. I think $s = ababab$ might shed some light
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:21
$begingroup$
Eh, I somehow missed the $ langle $, and its pair, don't ask me how ... And sure thing. :)
$endgroup$
– Andrew
Mar 18 '17 at 22:25
|
show 1 more comment
$begingroup$
Yeah, I saw that happening. I wonder if it's ever normal.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:11
$begingroup$
@FruitfulApproach Now that I think about it, subgroups share the parent's identity, so "", the empty string should be in $H$, but I think the empty string should also be a substring of every string... If so, then H isn't even a subgroup, because the identity isn't in it. :/ The topic seems interesting enough, though!
$endgroup$
– Andrew
Mar 18 '17 at 22:19
$begingroup$
It's called you "artificialy add it in", as it's trivial enough to do. Actually, $epsilon in H$ automatically, by definition of generated subgroup.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:20
$begingroup$
I'm making an answer. I think $s = ababab$ might shed some light
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:21
$begingroup$
Eh, I somehow missed the $ langle $, and its pair, don't ask me how ... And sure thing. :)
$endgroup$
– Andrew
Mar 18 '17 at 22:25
$begingroup$
Yeah, I saw that happening. I wonder if it's ever normal.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:11
$begingroup$
Yeah, I saw that happening. I wonder if it's ever normal.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:11
$begingroup$
@FruitfulApproach Now that I think about it, subgroups share the parent's identity, so "", the empty string should be in $H$, but I think the empty string should also be a substring of every string... If so, then H isn't even a subgroup, because the identity isn't in it. :/ The topic seems interesting enough, though!
$endgroup$
– Andrew
Mar 18 '17 at 22:19
$begingroup$
@FruitfulApproach Now that I think about it, subgroups share the parent's identity, so "", the empty string should be in $H$, but I think the empty string should also be a substring of every string... If so, then H isn't even a subgroup, because the identity isn't in it. :/ The topic seems interesting enough, though!
$endgroup$
– Andrew
Mar 18 '17 at 22:19
$begingroup$
It's called you "artificialy add it in", as it's trivial enough to do. Actually, $epsilon in H$ automatically, by definition of generated subgroup.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:20
$begingroup$
It's called you "artificialy add it in", as it's trivial enough to do. Actually, $epsilon in H$ automatically, by definition of generated subgroup.
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:20
$begingroup$
I'm making an answer. I think $s = ababab$ might shed some light
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:21
$begingroup$
I'm making an answer. I think $s = ababab$ might shed some light
$endgroup$
– Hermit with Adjoint
Mar 18 '17 at 22:21
$begingroup$
Eh, I somehow missed the $ langle $, and its pair, don't ask me how ... And sure thing. :)
$endgroup$
– Andrew
Mar 18 '17 at 22:25
$begingroup$
Eh, I somehow missed the $ langle $, and its pair, don't ask me how ... And sure thing. :)
$endgroup$
– Andrew
Mar 18 '17 at 22:25
|
show 1 more comment
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