text: # Mapping Question
  
  1. Apr 11, 2010
  
  ### Phred Willard
  
  Can an element of a quotient group G/N in an isomorphism f:G/N--H map back onto itself if it does not have a corresponding element in H? The example I am looking at is the quotient group of the group of symmetries of an n-gon, G, where n is an even number and N equal to the normal subgroup containing the identity and a 180 rotation. H represents another n-gon with an even n, but is less than the order of G.
  
  I have found that the quotient group G/N contains every permutation in G (which I think is incorrect) because for f to be an isomorphism, G contains rotations that don't map to H.
  
  What's going on here?
  
  2. Apr 12, 2010
  
  ### rs1n
  
  Are you referring to one of the isomorphism theorems?
  
  Let's do a small example, with the dihedral group $D_4$ (the group of symmetries of a square). The 8 elements are $\{ e, h, v, d_1, d_2, r_{90}, r_{180}, r_{270} \}$. If $N=\{ e, r_{180} \}$ then what are the elements of $D_4/N$? (From Lagrange's theorem, we expect only 4 elements in this group.)
  
  $eN = r_{180}N = N$
  $vN = hN = \{ h, v \}$
  $d_1N = d_2N = \{ d_1, d_2 \}$
  $r_{90}N = r_{270}N = \{ r_{90}, r_{270} \}$
  
  Notice that the elements of $D_4/N$ are essentially _cosets_ of N; so I am not sure what you mean by "G/N contains every permutation."
explanation: The text extract is a discussion about a specific problem in group theory, specifically dealing with quotient groups and isomorphisms. The user, Phred Willard, poses a question about whether an element of a quotient group \( G/N \) can map back onto itself if it does not have a corresponding element in \( H \). The user also mentions an example involving the group of symmetries of an n-gon and a normal subgroup containing the identity and a 180-degree rotation. The user is confused about the elements of the quotient group and their mapping under an isomorphism.
  
  Another user, rs1n, responds by providing a detailed example using the dihedral group \( D_4 \) to clarify the concept of quotient groups and cosets.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Mapping Question
  
  1. Apr 11, 2010
  
  ### Phred Willard
  
  Can an element of a quotient group G/N in an isomorphism f:G/N--H map back onto itself if it does not have a corresponding element in H? The example I am looking at is the quotient group of the group of symmetries of an n-gon, G, where n is an even number and N equal to the normal subgroup containing the identity and a 180 rotation. H represents another n-gon with an even n, but is less than the order of G.
  
  I have found that the quotient group G/N contains every permutation in G (which I think is incorrect) because for f to be an isomorphism, G contains rotations that don't map to H.
  
  What's going on here?
  
  2. Apr 12, 2010
  
  ### rs1n
  
  Are you referring to one of the isomorphism theorems?
  
  Let's do a small example, with the dihedral group $D_4$ (the group of symmetries of a square). The 8 elements are $\{ e, h, v, d_1, d_2, r_{90}, r_{180}, r_{270} \}$. If $N=\{ e, r_{180} \}$ then what are the elements of $D_4/N$? (From Lagrange's theorem, we expect only 4 elements in this group.)
  
  $eN = r_{180}N = N$
  $vN = hN = \{ h, v \}$
  $d_1N = d_2N = \{ d_1, d_2 \}$
  $r_{90}N = r_{270}N = \{ r_{90}, r_{270} \}$
  
  Notice that the elements of $D_4/N$ are essentially _cosets_ of N; so I am not sure what you mean by "G/N contains every permutation."
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is a discussion about a specific problem in group theory, specifically dealing with quotient groups and isomorphisms. The user, Phred Willard, poses a question about whether an element of a quotient group \( G/N \) can map back onto itself if it does not have a corresponding element in \( H \). The user also mentions an example involving the group of symmetries of an n-gon and a normal subgroup containing the identity and a 180-degree rotation. The user is confused about the elements of the quotient group and their mapping under an isomorphism.
  
  Another user, rs1n, responds by providing a detailed example using the dihedral group \( D_4 \) to clarify the concept of quotient groups and cosets.
  
  Label: question-and-answer