Proving Isomorphism of G to Subgroup of G/H + G/K

[tyrannosaurus]

Homework Statement

Let H and K be normal Subgroups of a group G s.t H intersect K = {e}. Show that G is isomorphic to a subgroup of G/H + G/K.

Homework Equations

G/H+G/K= direct product of G/H and G/K.

The Attempt at a Solution

Proof/
Lets define are mapping f:G to G/H+G/K by f(g)=(gH,GK). Need to show that this is a homomorphism. Obviously f is a well defined function (Do I need any more explanation here?). Operation Preserving- Let x,y be elements of G. Then f(xy)= (xyH,xyK)= (xHyH,xKyK)= (xH,XK),(yH,yK)=f(x)f(y) ( I need help justifying my step on this)
Thus f is a homomorphism. Since kerf ={g element of G| (gH,gK)=(H,K)}, then ker f={e}. Thus f is injective (one to one).
So By 1st isomorphism theorem, G/Ker f is isomorphic to f(G). Since f(G) is a subgroup of G/H+G/K (since f is one to one), then G is isomorphic to a subgroup of G/H + G/K (since G/Kerf= G because kerf={e}, is this right?)
My question is for clarification, does this seem like I have proven that G is isomorphic to a subgroup of G/H + G/K, or I have I proven that G is isomorphic to G/H + G/K ?

 

[Office_Shredder]

Your proof looks good. A general rule of thumb for well-definedness is that you only need to be concerned when your domain involves picking representative elements of cosets (and then whether the function is independent of representative). In this case you're not, only the codomain involves cosets so you're OK.

For the homomorphism part, where are you stuck? Let's look at it slowly

f(xy)= (xyH,xyK) This is how f is defined

= (xHyH,xKyK) This is by definition of multiplication by cosets

= (xH,XK),(yH,yK) This is by the definition of multiplication for the product of two groups

=f(x)f(y) this is by the definition of f again

Then once you have that kerf is trivial, G/kerf = G is isomorphic to the image of f by the first isomorphism theorem. As to whether G is isomorphic to G/H + G/K, this will only be true if the image of f is all of G/H + G/K. Is this always true?

 

[tyrannosaurus]

Your proof looks good. A general rule of thumb for well-definedness is that you only need to be concerned when your domain involves picking representative elements of cosets (and then whether the function is independent of representative). In this case you're not, only the codomain involves cosets so you're OK.

For the homomorphism part, where are you stuck? Let's look at it slowly

f(xy)= (xyH,xyK) This is how f is defined

= (xHyH,xKyK) This is by definition of multiplication by cosets

= (xH,XK),(yH,yK) This is by the definition of multiplication for the product of two groups

=f(x)f(y) this is by the definition of f again

Then once you have that kerf is trivial, G/kerf = G is isomorphic to the image of f by the first isomorphism theorem. As to whether G is isomorphic to G/H + G/K, this will only be true if the image of f is all of G/H + G/K. Is this always true?

Thanks for the help, I just needed some clarification.