Find all homomorphic images of the octic group D4Solutionr

Find all homomorphic images of the octic group D₄.

Solution

[r = rotation, s = reflection]

{1, r^2}
{1, r, r^2, r^3}
{1, r^2, s, sr^2}
{1, r^2, sr, sr^3}

call the subgroups H,K,M and N (we should also probably include the trivial normal subgroups {1} and D4).

find D4/D4 {1}, D4/H, D4/K, D4/M, D4/N and D4/{1} = D4.

here\'s what you know: K,L, and M are of order 4, so D4/K, D4/M and D4/N are of order 2. that makes describing D4/K, D4/M and D4/N easy.

so the only really \"interesting example\" is D4/H, which is of order 4. there are only 2 group types of order 4.

alculate D4/H. i\'ll get you started:

H = {1, r^2}. so here are the 4 cosets:

H
Hr = {r,r^3}
Hs = {s, r^2s} = {s,sr^2}
Hsr = {sr, r^2sr} = {sr,sr^3}

so D4/H = {H,Hr,Hs,Hsr}.

now...make a multiplication table. is D4/H abelian? is it cyclic? what are the orders of each coset?