Some basic algebra (using Isomorphism Theorems)

[bham10246]

Homework Statement

Let [itex]G[/itex] be a group with a normal subgroup [itex]N[/itex] and subgroups [itex] K \triangleleft H \leq G. [/itex]

If [itex]H/K [/itex] is nontrivial, prove that at least one of [itex]HN/KN[/itex] and [itex](H\cap N)/(K\cap N)[/itex] must be nontrivial.

Homework Equations

The Three (or Four) Isomorphism Theorems.

The Attempt at a Solution

By the first isomorphism theorem, we saw that [itex]HN/KN \cong H/K [/itex]. So if [itex]H/K[/itex] is nontrivial, then [itex]HN/KN [/itex] is also nontrivial.

Now to show that [itex](H\cap N)/(K\cap N)[/itex] is also nontrivial, what normal subgroup of [itex]H/K [/itex] is this quotient group [itex](H\cap N)/(K\cap N)[/itex] isomorphic to?

Because of the "and" in the statement of the problem, should both be nontrivial?

 

[CompuChip]

The problem asks you to prove that at least one of those two is non-trivial. If I read your post correctly, you've proven that the first one is always non trivial (given the err.. given data), so automatically at least one of them is. In which case, though, the question seems rather pointless (you've kind of reduced it to "show that if A always holds, then at least one of A or B holds"). But I don't know if your first steps are correct, since I don't know the isomorphism theorems

 

[Kummer]

2nd Isomorphism Theorem:
[tex]H/(H\cap K)\simeq HK/K[/tex] and [tex]H/(H\cap N)\simeq HN/N[/tex]

Form factor groups,
[tex](H/(H\cap K))/(H/(H\cap N))\simeq (HK/K)/(HN/N)[/tex]

3rd Isomorphism Theorem:
[tex](H\cap N)/(H\cap K)\simeq (HK/K)/(HN/N)[/tex]

Are you sure you are looking for,
[tex](H\cap N)/(K\cap N)[/tex]
And not for,
[tex](H\cap N)/(H\cap K)[/tex]
Because we do not know if [tex]K\leq N[/tex] and so cannot use theorem.

 

[bham10246]

Hi CompuChip, yes, your reasoning is correct and this part of the problem doesn't really make sense. But this is how I proved the first part of this problem which states: prove that [itex]HN/KN[/itex] is isomorphic with a quotient group of [itex]H/K[/itex].

So let [itex]fN\rightarrow H/K[/itex] be a homomorphism where [itex]f: hn\longmapsto hK [/itex] where [itex]h \in H, n \in N[/itex]. So [itex]f(hn)=K \Rightarrow h \in K \Rightarrow hn\in KN[/itex]. This shows that [itex]HN/KN \cong H/K[/itex]. So am I right?

And Kummer, I'm not sure... but isn't [itex]H\cap K = K[/itex]?