Maths Problem

Space Gap

Hi all,
I was just wondering if anyone could help with a maths question I have.
The question is

Let N1 and N2 be normal subgroups of a group G. Show that N1n N2 is also a normal subgroup of G.
Could anyone throw some light on how I even start it.
Thanks

Joey the lips

http://resources.metapress.com/pdf-preview.axd?code=j626r32627177801&size=smaller

perhaps...

Fremen

The way to deal with questions like that is first to ask yourself "what's the definition of a normal subgroup?".Now ask yourself "what do I know?" and "what do I want to show?".

I'm a bit reluctant to do too much of this for you. It's much better for your understanding if you do it yourself.

So, the starting point: If G is a group, what's the definition of a normal subgroup of N of G?
Post back and we'll take it from there.

Space Gap

Fremen wrote: »

The way to deal with questions like that is first to ask yourself "what's the definition of a normal subgroup?".Now ask yourself "what do I know?" and "what do I want to show?".

I'm a bit reluctant to do too much of this for you. It's much better for your understanding if you do it yourself.

So, the starting point: If G is a group, what's the definition of a normal subgroup of N of G?
Post back and we'll take it from there.

Hi Fremen,
What I know about a normal subgroup is that. A subgroup of N of G is normal in G, if and only if gH = Hg for all g elements in G.
Am I on the right track with this.

equivariant

Space Gap wrote: »

Hi Fremen,
What I know about a normal subgroup is that. A subgroup of N of G is normal in G, if and only if gH = Hg for all g elements in G.
Am I on the right track with this.

(You have used N and H for the same object)

Probably better to think element wise. In other words, a subgroup H of G is normal if and only if, for all h in H and g in G, g^(-1)hg is an element of H.

This is equivalent to the definition that you gave above (why?), but it is perhaps easier to work with this one for the problem that you are considering.

Fremen

Hi Space gap

Equivariant's approach is the right one to take.
Check out the definition on Wikipedia:
http://en.wikipedia.org/wiki/Normal_subgroup#Definitions

So you have two normal subgroups N1 and N2. What do elements of the product (N1)(N2) look like?

equivariant

Fremen wrote: »

Hi Space gap

Equivariant's approach is the right one to take.
Check out the definition on Wikipedia:
http://en.wikipedia.org/wiki/Normal_subgroup#Definitions

So you have two normal subgroups N1 and N2. What do elements of the product (N1)(N2) look like?

Is the original question about the intersection or the product? I thought it was intersection (about the same level of difficulty anyway)

ray giraffe

Space Gap wrote: »

Hi all,
I was just wondering if anyone could help with a maths question I have.
The question is

Let N1 and N2 be normal subgroups of a group G. Show that N1n N2 is also a normal subgroup of G.
Could anyone throw some light on how I even start it.
Thanks

Don't forget you have two steps!

(i) Show the intersection is a subgroup.
(ii) Show that the intersection is a normal subgroup.

The first step follows from the fact that "the intersection of 2 subgroups is a subgroup". The proof of that fact is a few lines long - do this first!

The second step is easy if equivariant's definition of "normal subgroup" is used, proof is a couple of lines.

Aside: Doing these proofs is a matter of meditating on the definitions. However the definition of normal subgroup is very strange when you first see it.

Ask your lecturer for examples of small groups (e.g. only 6 elements) with subgroups, some normal, some not normal.

Fremen

equivariant wrote: »

Is the original question about the intersection or the product? I thought it was intersection (about the same level of difficulty anyway)

Oh, good point. I assumed the n was a typo...

Space Gap

Thanks for all the help. I got it now.