Abstract Algebra (Normal Subgroups)

  • Thread starter mattmns
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  • #1
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Hello. We got a review today in abstract algebra, and I am stuck on two problems.

1) Let f: G -> H be a surjective homomorphism of groups. Prove that if K is a normal subgroup of G, then f(K) is a normal subgroup of H. Where f(K)= {f(k): k [tex]\in[/tex]K}
The entire f(K) part is really throwing me off.
We know:
G and H are groups
K is a normal subgroup of G
Also H is a normal subgroup because im_f = H (because f is surjective)
The real problem is I don't quite understand the whole f(K) part. Any ideas about what f(K) is, and any ideas about the problem?




2) If H and K are normal subgroups of a group G satisfying H[tex]\cap[/tex]K = {1}, prove that hk = kh for all h[tex]\in[/tex]H and k[tex]\in[/tex]K.
This one is really throwing me off.
We Know:
H and K are normal subgroups of a group G.
H[tex]\cap[/tex]K = {1} is a normal subgroup of G. (because of a previous problem I had proved).
Also, HK is a subgroup (by a theorem from my book)
For this problem it seems that it would be sufficient to show that G is abelian, but I am not sure how we would do that. Or maybe just use some general properties algebraically to show that hk=kh. I am not sure how we are supposed to use the H[tex]\cap[/tex]K = {1} though.
Any ideas about this one?




Thanks!!!
 
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Answers and Replies

  • #2
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subgroup test comes to mind. oh wait never mind you have to prove normal subgroup sorry prolly not that easy then
 
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  • #3
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Yes, but I am not understanding how I would say that 1 is in f(K), or x,y in f(K) => xy in f(K), my real problem is understanding how I would use f(K) to see if the properties of a subgroup apply to f(K). Any ideas about f(K)?
 
  • #4
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yes I had a problem like this but it was dealing with isomorphisms I think I assumed f(a) and f(b) were in f(k) then I said f(a)f(b) was in f(k) since ab is in k so f(ab) is in f(k) and f(ab)=f(a)f(b), similar argument holds for a inverse

f(k) is non empty because f(e) is in f(k)

since k is a group it has all the properties we need like identity and a,b in k implies ab in k so f(ab) is in f(k)
a in k then a^-1 in k so f(a^-1) is f(k)

f(a^-1) = (f(a))^-1
 
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  • #5
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Hmm, that is an interesting idea, I think I remember my teacher doing something similar, I will try it out, thanks. Any ideas for #2?
 
  • #6
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for the 1st one you've got to show that the subgroup f(K) is normal in H, that is, [tex]f(K) = f(x)f(K)f(x)^{-1}[/tex] for any f(x) in H if K is normal in G.
unless i made a mistake (or i'm getting better at algebra) it's not very hard. all you've got to use, after all, is properties of homomorphisms to show that the image is a normal subgroup. i think the reason you need a surjective homomorphism is so that you get a group in the image, otherwise you could get the empty set as an image which isn't a normal subgroup.
for the 2nd one i'd try out some examples. the quaternions is a noncommutative group in which every subgroup is normal, so you're idea won't work (showing it's abelian). try playing with a concrete example like the quaternions & see if that helps.
 
  • #7
AKG
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mattmns said:
Hello. We got a review today in abstract algebra, and I am stuck on two problems.

1) Let f: G -> H be a surjective homomorphism of groups. Prove that if K is a normal subgroup of G, then f(K) is a normal subgroup of H. Where f(K)= {f(k): k in K}
Suppose x is in f(K) and h is in H. To show f(K) is normal, you want to show that hxh-1 is in f(K). Well you know that since f is surjective, there is some g in G such that f(g) = h. Since x is in f(K), you know there is some k in K such that f(k) = x. So you can pick g and k such that:

hxh-1 = ... (figure the rest out)
2) If H and K are normal subgroups of a group G satisfying H[itex]\cap[/itex]K = {1}, prove that hk = kh for all h in H and k in K.
This one is really throwing me off.
Do you know that if H is normal, then for every g in G, gH = Hg? In other words, for every g in G and for every h in H, there is some h' in H (possibly the same as h) such that gh = h'g. This holds for every g in G, including every g in K. Can you go from here?
 
  • #8
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yeah i think i made a mistake. i should think harder
 
  • #9
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Thanks everyone, I got them both :smile:
 

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