Mapping Elements in Quotient Group G/N to Isomorphism f:G/N--H

[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?

 

[rs1n]

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?

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."