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Determining whether a relation disrupts freeness of a subgroup
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Suppose we examine a group with presentation $G = langle a, b, c ~vert~ c^2 rangle$. It's my suspicion that the subgroup $langle a, b rangle$ is free, since the relation in the presentation does not "touch" these generators. However, in more complicated scenarios, this may be difficult to determine. For example, if $G= langle a,b,c,d ~vert~ acd^-1b^-1 rangle$, is the subgroup $langle a, b rangle$ free? Again, I suspect yes, though I am not sure how to prove it, if it is even true.
Are there conditions on the relations that can help easily determine whether a subset of generators generates a free subgroup under the relations?
abstract-algebra group-theory
asked Jul 18 at 16:54
MightyTyGuy
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Suppose we examine a group with presentation $G = langle a, b, c ~vert~ c^2 rangle$. It's my suspicion that the subgroup $langle a, b rangle$ is free, since the relation in the presentation does not "touch" these generators. However, in more complicated scenarios, this may be difficult to determine. For example, if $G= langle a,b,c,d ~vert~ acd^-1b^-1 rangle$, is the subgroup $langle a, b rangle$ free? Again, I suspect yes, though I am not sure how to prove it, if it is even true.
Are there conditions on the relations that can help easily determine whether a subset of generators generates a free subgroup under the relations?
abstract-algebra group-theory
asked Jul 18 at 16:54
MightyTyGuy
1,020314
3
en.m.wikipedia.org/wiki/Freiheitssatz
– Steve D
Jul 18 at 18:30
add a comment |Â
up vote
1
down vote
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up vote
1
down vote
favorite
Suppose we examine a group with presentation $G = langle a, b, c ~vert~ c^2 rangle$. It's my suspicion that the subgroup $langle a, b rangle$ is free, since the relation in the presentation does not "touch" these generators. However, in more complicated scenarios, this may be difficult to determine. For example, if $G= langle a,b,c,d ~vert~ acd^-1b^-1 rangle$, is the subgroup $langle a, b rangle$ free? Again, I suspect yes, though I am not sure how to prove it, if it is even true.
Are there conditions on the relations that can help easily determine whether a subset of generators generates a free subgroup under the relations?
abstract-algebra group-theory
asked Jul 18 at 16:54
MightyTyGuy
1,020314
Suppose we examine a group with presentation $G = langle a, b, c ~vert~ c^2 rangle$. It's my suspicion that the subgroup $langle a, b rangle$ is free, since the relation in the presentation does not "touch" these generators. However, in more complicated scenarios, this may be difficult to determine. For example, if $G= langle a,b,c,d ~vert~ acd^-1b^-1 rangle$, is the subgroup $langle a, b rangle$ free? Again, I suspect yes, though I am not sure how to prove it, if it is even true.
Are there conditions on the relations that can help easily determine whether a subset of generators generates a free subgroup under the relations?
abstract-algebra group-theory
asked Jul 18 at 16:54
MightyTyGuy
1,020314
asked Jul 18 at 16:54
MightyTyGuy
1,020314
asked Jul 18 at 16:54
MightyTyGuy
1,020314
asked Jul 18 at 16:54
MightyTyGuy
1,020314
1,020314
3
en.m.wikipedia.org/wiki/Freiheitssatz
– Steve D
Jul 18 at 18:30
add a comment |Â
3
en.m.wikipedia.org/wiki/Freiheitssatz
– Steve D
Jul 18 at 18:30
3
3
en.m.wikipedia.org/wiki/Freiheitssatz
– Steve D
Jul 18 at 18:30
en.m.wikipedia.org/wiki/Freiheitssatz
– Steve D
Jul 18 at 18:30
add a comment |Â
1 Answer
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The first example is a free product of the free group on $langle a,b rangle$ and the cyclic group $langle c rangle$ of order $2$.
In the second example, you just eliminate the generator $d$, and see that $G$ is the free group on $a,b,c$.
answered Jul 18 at 16:59
Derek Holt
49.7k53366
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
The first example is a free product of the free group on $langle a,b rangle$ and the cyclic group $langle c rangle$ of order $2$.
In the second example, you just eliminate the generator $d$, and see that $G$ is the free group on $a,b,c$.
answered Jul 18 at 16:59
Derek Holt
49.7k53366
add a comment |Â
up vote
3
down vote
The first example is a free product of the free group on $langle a,b rangle$ and the cyclic group $langle c rangle$ of order $2$.
In the second example, you just eliminate the generator $d$, and see that $G$ is the free group on $a,b,c$.
answered Jul 18 at 16:59
Derek Holt
49.7k53366
add a comment |Â
up vote
3
down vote
up vote
3
down vote
The first example is a free product of the free group on $langle a,b rangle$ and the cyclic group $langle c rangle$ of order $2$.
In the second example, you just eliminate the generator $d$, and see that $G$ is the free group on $a,b,c$.
answered Jul 18 at 16:59
Derek Holt
49.7k53366
The first example is a free product of the free group on $langle a,b rangle$ and the cyclic group $langle c rangle$ of order $2$.
In the second example, you just eliminate the generator $d$, and see that $G$ is the free group on $a,b,c$.
answered Jul 18 at 16:59
Derek Holt
49.7k53366
answered Jul 18 at 16:59
Derek Holt
49.7k53366
answered Jul 18 at 16:59
Derek Holt
49.7k53366
answered Jul 18 at 16:59
Derek Holt
49.7k53366
49.7k53366
add a comment |Â
add a comment |Â
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3
en.m.wikipedia.org/wiki/Freiheitssatz
– Steve D
Jul 18 at 18:30