Suppose that G is a group satisfying the Ascending Chain Condition (ACC) and the Descending Chain Condition (DCC) on normal groups.
Show that if H,J,K are groups such that G is isomorphic to the direct product HxJ of H and J, and G is isomorphic to HxK, then J,K are isomorphic.
Since H,J,K must satisfy the ACC and DCC on normal subgroups, this exercise seems to require a straightforward application of the Krull-Schmidt Theorem, but I don't see how to do this, in other words, I don't know that theorem. Anyone have any ideas ?
ACC and DCC
Bắt đầu bởi nemo, 14-10-2005 - 16:42
#1
Đã gửi 14-10-2005 - 16:42
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#2
Đã gửi 15-10-2005 - 09:29
The Krull-Schmidt Theorem states that if http://dientuvietnam...n/mimetex.cgi?G is a group satisfying both the ACC and DCC conditions on normal supgroups then it decomposes into direct product of a finite number of indecomposable groups. This decomposition is unique up to isomorphism and permutation.
Knowing that http://dientuvietnam.../mimetex.cgi?G. I think now it's not difficult to conclude that http://dientuvietnam...n/mimetex.cgi?J and http://dientuvietnam...n/mimetex.cgi?K are isomorphic using the uniqueness property mentioned in the Krull-Schmidt Theorem.
For an acount of ACC and DCC conditions and the Krull-Schmidt Theorem, see Hungerford's book (Algebra, GTM).
Knowing that http://dientuvietnam.../mimetex.cgi?G. I think now it's not difficult to conclude that http://dientuvietnam...n/mimetex.cgi?J and http://dientuvietnam...n/mimetex.cgi?K are isomorphic using the uniqueness property mentioned in the Krull-Schmidt Theorem.
For an acount of ACC and DCC conditions and the Krull-Schmidt Theorem, see Hungerford's book (Algebra, GTM).
<span style='color:blue'>Thu đi để lại lá vàng
Anh đi để lại cho nàng thằng ku</span>
Anh đi để lại cho nàng thằng ku</span>
#3
Đã gửi 15-10-2005 - 13:04
Yes, I am going to do it soon, I just has that book. Thanks so much !For an acount of ACC and DCC conditions and the Krull-Schmidt Theorem, see Hungerford's book (Algebra, GTM).
The Krull-Schmidt Theorem states that if is a group satisfying both the ACC and DCC conditions on normal supgroups then it decomposes into direct product of a finite number of indecomposable groups. This decomposition is unique up to isomorphism and permutation
Thus, Krull-Schmidt is the main idea.
Part of the problem though is that http://dientuvietnam...mimetex.cgi?HxJ need not be equal to http://dientuvietnam...imetex.cgi?HxK; they simply need to be isomorphic. If we let H have indecomposable factors http://dientuvietnam...gi?H_1,...,H_m. J have indecomposable factors http://dientuvietnam...cgi?J_1,...,J_n, and K have factors http://dientuvietnam...cgi?K_1,...,K_p, then we can conclude that p=n. Let http://dientuvietnam...mimetex.cgi?i_1 be injection of H into http://dientuvietnam...mimetex.cgi?HxK and http://dientuvietnam...mimetex.cgi?i_2 be injection of K into http://dientuvietnam...imetex.cgi?HxK. Let f be the isomorphism from http://dientuvietnam...mimetex.cgi?HxK to http://dientuvietnam.net/cgi-bin/mimetex.cgi?HxJ. Then http://dientuvietnam.net/cgi-bin/mimetex.cgi?G=HxJ=f(HxK)=f(i_1(H))xf(i_2(K)). One can conclude that http://dientuvietnam.net/cgi-bin/mimetex.cgi?H_r and http://dientuvietnam.net/cgi-bin/mimetex.cgi?f(i_1(H_r)) are isomorphic for , however, it is not obvious to me how this helps my cause. Any thoughts?
p/s: What is GTM ?
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#4
Đã gửi 15-10-2005 - 14:59
GTM = Graduate Texts in Mathematics
A\times Bhttp://dientuvietnam.net/cgi-bin/mimetex.cgi?m.
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#5
Đã gửi 16-10-2005 - 21:28
I think you can proceed by induction on .
Yes, I see.
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#6
Đã gửi 18-10-2005 - 15:09
Remark: This Krull - Schmidt theorem for groups is likely to the Jordan - Holder theorem, which is for finite lenth modules:
If a module M is finite lendth then every "composition series" of M has the same lenth and the elements in the composition series are some permutation of the elements in an arbitrary choosen composition series.
If a module M is finite lendth then every "composition series" of M has the same lenth and the elements in the composition series are some permutation of the elements in an arbitrary choosen composition series.
Lạy chúa!
Con không hề hoài nghi tí nào về sự hiện hữu hoài nghi của người nhưng con hoài nghi rất nhiều về sự minh mẫn và công bình của người!
Con không hề hoài nghi tí nào về sự hiện hữu hoài nghi của người nhưng con hoài nghi rất nhiều về sự minh mẫn và công bình của người!
#7
Đã gửi 20-10-2005 - 09:13
To have a connection with the Jordan - Holder theorem, I need a reference for Jordan-Holder Theorem for topological groups. I tried to prove it by imitating the proof of Jordan-Holder Theorem for R-Modules but I found I should assume that the locally and segma-compact to use the third isomorphism theorem which is only hold under this condition for the topological groups.
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