# Thread: groups...normalizer of subgroup..

1. ## groups...normalizer of subgroup..

if H is a subgroup of G..
let N(H)={g belongs to G/ gHg^=H} , show that
1) N(H) is a subgroup of G,
2) H is normal in N(H) ,
3) if H is normal in G iff N(H)=G,
4) if H is a normal subgroup of the subgroup K in G, then K is a subset of or = N(H)

how do we prove? im a beginner in groups...any help would be greatful..

2. Originally Posted by kashuv
if H is a subgroup of G..
let N(H)={g belongs to G/ gHg^=H} , show that
1) N(H) is a subgroup of G,
2) H is normal in N(H) ,
3) if H is normal in G iff N(H)=G,
4) if H is a normal subgroup of the subgroup K in G, then K is a subset of or = N(H)

how do we prove? im a beginner in groups...any help would be greatful..
you cannot be that beginner if you're in normalizers of subgroups...at least 3-4 hours of class?

Anyway, what's the definition of subgroup? And how do we check whether a given set is a subgroup of a given group? Begin with this and try to prove (1), and now apply the definition of normal subgroup of a group to prove (2) and (3).
(4) is, perhaps, the "hardest", as it actually asks from you to prove maximality of something, but let's first see if you can handle the above.

Tonio

3. hi ..
i managed to solve the 1st 3 by googling bout normaliser... 4th sub question is where im stuck at

4. Originally Posted by kashuv
hi ..
i managed to solve the 1st 3 by googling bout normaliser... 4th sub question is where im stuck at

Well, google a little more and read carefully the definition of normalizer of a subgroup H: it turns out that it is the

sbgp. of G which is maximal wrt the property that H is normal in it...

Tonio