How to obtain isomorphic factor groups of a group from its ismomorphic subgroups?

Dear Friends,

If two normal subgroups H and K of a group G are ısomorphic then the factor groups G\H and G\K need not be isomorphic. My question is what are further conditions we need on H, K and probably on G that gaurantee the isomorphisms of G\H and G\K when H and K are isomorphic

Re: How to obtain isomorphic factor groups of a group from its ismomorphic subgroups?

i claim the following is a sufficient (do not know about necessary) condition on H,K and G:

there exists σ in Aut(G) with σ(H) = K.

we would hope that this defines an isomorphism:

σ_{H/K} from G/H to G/K, given by:

σ_{H/K}(gH) = σ(g)K.

first, we need to verify that σ_{H/K} is well-defined.

suppose gH = g'H. then g'^{-1}g is in H, so σ(g'^{-1}g) = (σ(g'))^{-1}σ(g) is in K, so:

σ(g)K = σ(g')K, thus σ_{H/K}(gH) = σ_{H/K}(g'H).

is σ_{H/K} a homomorphism?

well, σ_{H/K}((gH)(g'H)) = σ_{H/K}(gg'H) = σ(gg')K = σ(g)σ(g')K = (σ(g)K)(σ(g')K) = σ_{H/K}(g)σ_{H/K}(g').

is σ_{H/K} injective?

suppose σ_{H/K}(gH) = σ(g)K = K. then σ(g) is in K, so g = σ^{-1}(σ(g)) is in σ^{-1}(K) = H.

thus the the only coset of H that is in the kernel of σ_{H/K} is H.

is σ_{H/K} surjective?

suppose that we have ANY coset of K in G/K, say xK. let g = σ^{-1}(x). then σ_{H/K}(gH) = σ(g)K = σ(σ^{-1}(x))K = xK.

ta da!