Symmetry group of equilateral triangle
up vote
1
down vote
favorite
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 at 9:17
Chetan Waghela
637
add a comment |
up vote
1
down vote
favorite
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 at 9:17
Chetan Waghela
637
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
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 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 at 9:17
Chetan Waghela
637
asked Nov 21 at 9:17
Chetan Waghela
637
asked Nov 21 at 9:17
Chetan Waghela
637
asked Nov 21 at 9:17
Chetan Waghela
637
asked Nov 21 at 9:17
Chetan Waghela
637
637
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
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 at 9:35
Dietrich Burde
76.9k64286
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 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 at 10:01
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
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 at 9:35
Dietrich Burde
76.9k64286
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 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 at 10:01
add a comment |
up vote
1
down vote
accepted
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 at 9:35
Dietrich Burde
76.9k64286
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 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 at 10:01
add a comment |
up vote
1
down vote
accepted
up vote
1
down vote
accepted
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 at 9:35
Dietrich Burde
76.9k64286
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 at 9:35
Dietrich Burde
76.9k64286
edited Nov 21 at 9:40
answered Nov 21 at 9:35
Dietrich Burde
76.9k64286
answered Nov 21 at 9:35
Dietrich Burde
76.9k64286
answered Nov 21 at 9:35
Dietrich Burde
76.9k64286
76.9k64286
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 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 at 10:01
add a comment |
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 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 at 10:01
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 at 9:51
I am not a mathematician what does the symbol in last sentence mean ?
– Chetan Waghela
Nov 21 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 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 at 10:01
add a comment |
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