Chủ đề này có 4 trả lời

[nemo]

We need more posts in this topic, I thinks

Now let's discuss the followings:

1) Let M and N be nilpotent subgroups of a group G. Give an example to show that when M and N are not both normal in G, the group MN need not be nilpotent.

2) Let M and N be solvable subgroups of a group G. Give an example to show that when both M and N are not normal in G, the group MN need not be solvable.

I expect answers from everyone !

[canh_dieu]

What is your definition of http://dientuvietnam.../mimetex.cgi?MN when both http://dientuvietnam...n/mimetex.cgi?M and http://dientuvietnam...n/mimetex.cgi?N are not normal subgroups of http://dientuvietnam.../mimetex.cgi?G?

[nemo]

Definition of MN is MN={mn | m M, n N}. We have, if H is a normal subgroup and K is a subgroup of a group G then HK be a subgroup of G but it is not equivalent (!?) (I am not sure ).

[quantum-cohomology]

Let M a compact complex Lie group, U a complex lie subgroup and biholomorph to http://dientuvietnam.net/cgi-bin/mimetex.cgi?\mathbb{C}^n. Question: Is U nilpotent?

Bài viết đã được chỉnh sửa nội dung bởi quantum-cohomology: 13-12-2005 - 00:03

[kakalot]

Like canh_dieu I don't know what MN will be when M and N are not both normal in G.
I just give an example :" Let M and N be nilpotent subgroups of a group G. Give an example to show that when M normal in G, N is not normal in G, the group MN need not be nilpotent"

We have known that S_3 is not nipoten because Z(S_3)=1.
for n 4 then 1, A_n are the only normal subgroups of S_n .
Let :
C(3)=<(123)>=A_3 then |C(3)|=3 then C(3) is nilpotent
C(2)=<(12)>= then |C(2)|=2 then C(2) is nilpotent and not normal in S_3.
Since C(3) C(2)=1 so we have S_3=C(2)C(3)