Mixing
Symmetry group of equilateral triangle
I have read at some places that the symmetry of equilateral triangle is C3v
as well as some places mention it to be D3.
The group tables for these two groups differ, hence they are not isomorphic.
Yet both these groups define symmetry of same shape.
Please, explain what is going on.
group-theory finite-groups symmetric-groups
asked Nov 21 '18 at 9:17
Chetan Waghela
637
add a comment |
I have read at some places that the symmetry of equilateral triangle is C3v
as well as some places mention it to be D3.
The group tables for these two groups differ, hence they are not isomorphic.
Yet both these groups define symmetry of same shape.
Please, explain what is going on.
group-theory finite-groups symmetric-groups
asked Nov 21 '18 at 9:17
Chetan Waghela
637
add a comment |
I have read at some places that the symmetry of equilateral triangle is C3v
as well as some places mention it to be D3.
The group tables for these two groups differ, hence they are not isomorphic.
Yet both these groups define symmetry of same shape.
Please, explain what is going on.
group-theory finite-groups symmetric-groups
asked Nov 21 '18 at 9:17
Chetan Waghela
637
I have read at some places that the symmetry of equilateral triangle is C3v
as well as some places mention it to be D3.
The group tables for these two groups differ, hence they are not isomorphic.
Yet both these groups define symmetry of same shape.
Please, explain what is going on.
group-theory finite-groups symmetric-groups
group-theory finite-groups symmetric-groups
asked Nov 21 '18 at 9:17
Chetan Waghela
637
asked Nov 21 '18 at 9:17
Chetan Waghela
637
asked Nov 21 '18 at 9:17
Chetan Waghela
637
asked Nov 21 '18 at 9:17
Chetan Waghela
637
asked Nov 21 '18 at 9:17
Chetan Waghela
637
637
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
The symmetry group of an equilateral triangle is the dihedral group $D_3$ with $6$ elements. It is a non-abelian group and hence isomorphic to $S_3$, since $C_6$ is abelian and there are only two different groups of order $6$. So there is one and only one symmetry group of the regular $3$-gon up to isomorphism. In particular, $C_{3v}cong D_3$.
Reference: see page $105$ here.
answered Nov 21 '18 at 9:35
Dietrich Burde
77.9k64386
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 '18 at 9:51
1
$Gcong H$ means that $G$ and $H$ are isomorphic as groups. So we may consider them as the same group (since you deal with irreducible representations of groups I assume that you are familiar with isomorphisms. It also seems that you never have accepted any answer:) ).
– Dietrich Burde
Nov 21 '18 at 10:01
add a comment |
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1 Answer
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1 Answer
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oldest
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The symmetry group of an equilateral triangle is the dihedral group $D_3$ with $6$ elements. It is a non-abelian group and hence isomorphic to $S_3$, since $C_6$ is abelian and there are only two different groups of order $6$. So there is one and only one symmetry group of the regular $3$-gon up to isomorphism. In particular, $C_{3v}cong D_3$.
Reference: see page $105$ here.
answered Nov 21 '18 at 9:35
Dietrich Burde
77.9k64386
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 '18 at 9:51
1
$Gcong H$ means that $G$ and $H$ are isomorphic as groups. So we may consider them as the same group (since you deal with irreducible representations of groups I assume that you are familiar with isomorphisms. It also seems that you never have accepted any answer:) ).
– Dietrich Burde
Nov 21 '18 at 10:01
add a comment |
The symmetry group of an equilateral triangle is the dihedral group $D_3$ with $6$ elements. It is a non-abelian group and hence isomorphic to $S_3$, since $C_6$ is abelian and there are only two different groups of order $6$. So there is one and only one symmetry group of the regular $3$-gon up to isomorphism. In particular, $C_{3v}cong D_3$.
Reference: see page $105$ here.
answered Nov 21 '18 at 9:35
Dietrich Burde
77.9k64386
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 '18 at 9:51
1
$Gcong H$ means that $G$ and $H$ are isomorphic as groups. So we may consider them as the same group (since you deal with irreducible representations of groups I assume that you are familiar with isomorphisms. It also seems that you never have accepted any answer:) ).
– Dietrich Burde
Nov 21 '18 at 10:01
add a comment |
The symmetry group of an equilateral triangle is the dihedral group $D_3$ with $6$ elements. It is a non-abelian group and hence isomorphic to $S_3$, since $C_6$ is abelian and there are only two different groups of order $6$. So there is one and only one symmetry group of the regular $3$-gon up to isomorphism. In particular, $C_{3v}cong D_3$.
Reference: see page $105$ here.
answered Nov 21 '18 at 9:35
Dietrich Burde
77.9k64386
The symmetry group of an equilateral triangle is the dihedral group $D_3$ with $6$ elements. It is a non-abelian group and hence isomorphic to $S_3$, since $C_6$ is abelian and there are only two different groups of order $6$. So there is one and only one symmetry group of the regular $3$-gon up to isomorphism. In particular, $C_{3v}cong D_3$.
Reference: see page $105$ here.
answered Nov 21 '18 at 9:35
Dietrich Burde
77.9k64386
edited Nov 21 '18 at 9:40
answered Nov 21 '18 at 9:35
Dietrich Burde
77.9k64386
answered Nov 21 '18 at 9:35
Dietrich Burde
77.9k64386
answered Nov 21 '18 at 9:35
Dietrich Burde
77.9k64386
77.9k64386
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 '18 at 9:51
1
$Gcong H$ means that $G$ and $H$ are isomorphic as groups. So we may consider them as the same group (since you deal with irreducible representations of groups I assume that you are familiar with isomorphisms. It also seems that you never have accepted any answer:) ).
– Dietrich Burde
Nov 21 '18 at 10:01
add a comment |
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 '18 at 9:51
1
$Gcong H$ means that $G$ and $H$ are isomorphic as groups. So we may consider them as the same group (since you deal with irreducible representations of groups I assume that you are familiar with isomorphisms. It also seems that you never have accepted any answer:) ).
– Dietrich Burde
Nov 21 '18 at 10:01
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 '18 at 9:51
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 '18 at 9:51
1
1
$Gcong H$ means that $G$ and $H$ are isomorphic as groups. So we may consider them as the same group (since you deal with irreducible representations of groups I assume that you are familiar with isomorphisms. It also seems that you never have accepted any answer:) ).
– Dietrich Burde
Nov 21 '18 at 10:01
$Gcong H$ means that $G$ and $H$ are isomorphic as groups. So we may consider them as the same group (since you deal with irreducible representations of groups I assume that you are familiar with isomorphisms. It also seems that you never have accepted any answer:) ).
– Dietrich Burde
Nov 21 '18 at 10:01
add a comment |
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