That doesn't make any sense, because [itex]G/(H \cap K)[/itex] is not a subgroup of [itex]G[/itex]. What you need is a homomorphism with kernel [itex]H \cap K[/itex].Avatarjoe said:Homework Statement
Suppose H and K are normal subgroups of G. Prove that G/H x G/K has a subgroup isomorphic to G/(H[itex]\cap[/itex]K)
The Attempt at a Solution
I was trying to find a homomorphism from G to G/H x G/K where G/(H[itex]\cap[/itex]K) is the kernal.
That doesn't make any sense, because G/(H∩K) is not a subgroup of G. What you need is a homomorphism with kernel H∩K.
Assuming you're working with right cosets, the natural homomorphism from G to G/H is g↦Hg, and the natural homomorphism from G to G/K is g↦Kg. Can you use these to construct a homomorphism from G to G/H x G/K?
No need to do that - surely you have encountered the theorem that the image of a homomorphism is always a subgroup. If not, it's easy to prove - easier than trying to show it for a specific case like this one.Avatarjoe said:If I'm not mistaken, I just need to show that {Hg x Kg} is a subgroup of G/H x G/K
micromass said:Another question: have you seen the isomorphism theorems yet?? They might come in handy.
A subgroup is a subset of a group that also forms a group under the same operation. In other words, a subgroup has all the same properties as the original group, but with a smaller set of elements.
G/H X G/K is the direct product of two quotient groups, where G is the original group and H and K are normal subgroups of G. This means that G/H X G/K is the set of all ordered pairs (gH, gK) where g is an element of G.
In order to prove isomorphism between two groups, we need to show that there exists a bijective homomorphism between them. This means that we need to find a function that maps elements from one group to the other in a way that preserves the group structure (i.e. the operation and identity element).
Proving isomorphism between two groups shows that they have the same structure and are essentially the same group, just with different elements. This can help us understand the properties and relationships between different groups.
Yes, this proof can be applied to any group and its subgroups. As long as the subgroups are normal and the group operation is well-defined, we can use this method to show isomorphism between a subgroup of the direct product of two quotient groups and the quotient group of the intersection of the subgroups.