# Thread: Homomorphic images of the dihedral group D4 of order 8

1. ## Homomorphic images of the dihedral group D4 of order 8

Find all homomorphic images of the dihedral group D4 of order 8, and give a homomorphism which produces each image. Give a familiar group that is isomorphic to each of the images.

2. ## Re: Homomorphic images of the dihedral group D4 of order 8

hint: by the first isomorphism theorem, any homomorphic image of D4 is isomorphic to a quotient group of D4.

so, first, find all the normal subgroups of D4. if memory serves me right, there are only 4 of them.

3. ## Re: Homomorphic images of the dihedral group D4 of order 8

[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}

i think its right, but i don't know what to do after this

4. ## Re: Homomorphic images of the dihedral group D4 of order 8

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.

5. ## Re: Homomorphic images of the dihedral group D4 of order 8

What is the homomorphic image then?

6. ## Re: Homomorphic images of the dihedral group D4 of order 8

calculate 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?

7. ## Re: Homomorphic images of the dihedral group D4 of order 8

i think it is both abelian and cyclic and orders are 1, and 3?
I'm confused

8. ## Re: Homomorphic images of the dihedral group D4 of order 8

yes, you are. how can any group of order 4 possibly have an element of order 3?????

if you think it is cyclic, prove it. which element has order 4?

9. ## Re: Homomorphic images of the dihedral group D4 of order 8

According to Lagrange's theorem every element must have an order that divides 4.
So, if we take any non-identity element, the order will be be 2 or 4.
If there is one of order 4, then the group is cyclic.

And all cyclic groups are abelian

10. ## Re: Homomorphic images of the dihedral group D4 of order 8

all of this is true, but you have yet to convince me that you have calculated ANY of the orders of {H,Hr,Hs,Hsr}.

you seem to be convinced this group has an element of order 4. i ask you: which one?

r r^3

12. ## Re: Homomorphic images of the dihedral group D4 of order 8

so you claim that Hr = {r,r^3} has order 4. hmm.

(Hr)(Hr) = Hr^2 = {r^2,r^4} = {r^2,1} = H.

it appears to me that Hr has order TWO.

,

,

,

,

,

,

,

,

,

,

,

,

,

,

# determine all homomorphic images of d8 pdf

Click on a term to search for related topics.