(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

2. Relevant equations

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

3. 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 ?

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# Homework Help: Isomorphism question

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