Examples of Polycyclic Group
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I'm reading about polycyclic groups recently.
Could anyone please give me some example of polycyclic groups?
group-theory
asked Nov 17 at 15:37
N3d4
102
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I'm reading about polycyclic groups recently.
Could anyone please give me some example of polycyclic groups?
group-theory
asked Nov 17 at 15:37
N3d4
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All finitely generated abelian groups and nilpotent groups are polycyclic.
– CyclotomicField
Nov 17 at 15:41
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up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm reading about polycyclic groups recently.
Could anyone please give me some example of polycyclic groups?
group-theory
asked Nov 17 at 15:37
N3d4
102
I'm reading about polycyclic groups recently.
Could anyone please give me some example of polycyclic groups?
group-theory
group-theory
asked Nov 17 at 15:37
N3d4
102
asked Nov 17 at 15:37
N3d4
102
asked Nov 17 at 15:37
N3d4
102
asked Nov 17 at 15:37
N3d4
102
asked Nov 17 at 15:37
N3d4
102
102
All finitely generated abelian groups and nilpotent groups are polycyclic.
– CyclotomicField
Nov 17 at 15:41
add a comment |
All finitely generated abelian groups and nilpotent groups are polycyclic.
– CyclotomicField
Nov 17 at 15:41
All finitely generated abelian groups and nilpotent groups are polycyclic.
– CyclotomicField
Nov 17 at 15:41
All finitely generated abelian groups and nilpotent groups are polycyclic.
– CyclotomicField
Nov 17 at 15:41
add a comment |
1 Answer
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Finitely generated nilpotent groups are polycyclic. This gives many examples. However, not every finitely generated solvable group is polycyclic. A well-known counterexamples is the Baumslag-Solitar group $BS(1,2)$.
The following result also may give some idea about polycyclic groups. Philip Hall conjectured, and Louis Auslander proved that every polycyclic group can be faithfully embedded into the integer unimodular group $SL_n(Bbb{Z})$ for some $n$. Conversely Anatoly Maltsev proved that solvable subgroups of $GL_n(Bbb{Z})$ are polycyclic.
Explicit examples: All dihedral groups $D_n$ with $n=2^k$ and the infinite dihedral group $D_{infty}$.
answered Nov 17 at 19:11
Dietrich Burde
76.8k64286
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Finitely generated nilpotent groups are polycyclic. This gives many examples. However, not every finitely generated solvable group is polycyclic. A well-known counterexamples is the Baumslag-Solitar group $BS(1,2)$.
The following result also may give some idea about polycyclic groups. Philip Hall conjectured, and Louis Auslander proved that every polycyclic group can be faithfully embedded into the integer unimodular group $SL_n(Bbb{Z})$ for some $n$. Conversely Anatoly Maltsev proved that solvable subgroups of $GL_n(Bbb{Z})$ are polycyclic.
Explicit examples: All dihedral groups $D_n$ with $n=2^k$ and the infinite dihedral group $D_{infty}$.
answered Nov 17 at 19:11
Dietrich Burde
76.8k64286
add a comment |
up vote
1
down vote
accepted
Finitely generated nilpotent groups are polycyclic. This gives many examples. However, not every finitely generated solvable group is polycyclic. A well-known counterexamples is the Baumslag-Solitar group $BS(1,2)$.
The following result also may give some idea about polycyclic groups. Philip Hall conjectured, and Louis Auslander proved that every polycyclic group can be faithfully embedded into the integer unimodular group $SL_n(Bbb{Z})$ for some $n$. Conversely Anatoly Maltsev proved that solvable subgroups of $GL_n(Bbb{Z})$ are polycyclic.
Explicit examples: All dihedral groups $D_n$ with $n=2^k$ and the infinite dihedral group $D_{infty}$.
answered Nov 17 at 19:11
Dietrich Burde
76.8k64286
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Finitely generated nilpotent groups are polycyclic. This gives many examples. However, not every finitely generated solvable group is polycyclic. A well-known counterexamples is the Baumslag-Solitar group $BS(1,2)$.
The following result also may give some idea about polycyclic groups. Philip Hall conjectured, and Louis Auslander proved that every polycyclic group can be faithfully embedded into the integer unimodular group $SL_n(Bbb{Z})$ for some $n$. Conversely Anatoly Maltsev proved that solvable subgroups of $GL_n(Bbb{Z})$ are polycyclic.
Explicit examples: All dihedral groups $D_n$ with $n=2^k$ and the infinite dihedral group $D_{infty}$.
answered Nov 17 at 19:11
Dietrich Burde
76.8k64286
Finitely generated nilpotent groups are polycyclic. This gives many examples. However, not every finitely generated solvable group is polycyclic. A well-known counterexamples is the Baumslag-Solitar group $BS(1,2)$.
The following result also may give some idea about polycyclic groups. Philip Hall conjectured, and Louis Auslander proved that every polycyclic group can be faithfully embedded into the integer unimodular group $SL_n(Bbb{Z})$ for some $n$. Conversely Anatoly Maltsev proved that solvable subgroups of $GL_n(Bbb{Z})$ are polycyclic.
Explicit examples: All dihedral groups $D_n$ with $n=2^k$ and the infinite dihedral group $D_{infty}$.
answered Nov 17 at 19:11
Dietrich Burde
76.8k64286
edited Nov 17 at 20:01
answered Nov 17 at 19:11
Dietrich Burde
76.8k64286
answered Nov 17 at 19:11
Dietrich Burde
76.8k64286
answered Nov 17 at 19:11
Dietrich Burde
76.8k64286
76.8k64286
add a comment |
add a comment |
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All finitely generated abelian groups and nilpotent groups are polycyclic.
– CyclotomicField
Nov 17 at 15:41