- #1
mattmns
- 1,128
- 6
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!
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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