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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!!!

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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