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November 4th, 2013, 02:03 AM   #1
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solvable Group of matrices over Z_p

show that the set of invertible upper triangular matrices over is a Group for matrix-multiplication.\\



so I showed that the set
is a group by showing the group-axioms. And the order of the Group is then
\item show that the set is a commutative normal subgroup of M .\\


that was also no problem ( just checking the subgroup-criterium+ normality)

\item ... but now comes my Problem : show that M is solvable by using this normal subgroup:\\
\vspace{0.5cm}

I get the series of normal subgroups: with the factors and .\\
is cyclic and so abelian. but my problem is ... if i could show that is abelian , then M would be solvable ... but how can i get that????





\end{enumerate}[/latex]
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November 4th, 2013, 04:19 AM   #2
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Re: solvable Group of matrices over Z_p

Actually, a general result holds true : The group of invertible upper-triangular martices over any finite field is solvable.

Try showing the finiteness of the derived series of M.
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November 4th, 2013, 04:58 AM   #3
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Re: solvable Group of matrices over Z_p

But if the Order of M = then this series has to be finite ... or am i completely on the wrong way ?

I wanted to show that with a theorem shown in our course : Let . If N and M/N are solveable, then M is solveable.

N is of course solveable because it was order = p . But M/N would have order = and how can i show that this is solveable ..
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November 4th, 2013, 06:14 AM   #4
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Re: solvable Group of matrices over Z_p

Quote:
Originally Posted by Sandra93
But if the Order of M = then this series has to be finite
Why do you think so? (NOTE* : I have not checked your calculation of order, but I think it is correct).

Quote:
Originally Posted by Sandra93
N is of course solvable because it was order = p.
Is p a prime?

Quote:
Originally Posted by Sandra93
But M/N would have order = and how can i show that this is solveable ..
I am not sure how to use the order to prove solvability of M/N. In fact, I am unsure if it can be approached this way. My argument using induction seems much more better, and I am not sure why you think that is unsatisfactory.
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November 4th, 2013, 09:43 AM   #5
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Re: solvable Group of matrices over Z_p

oh i forgot this part .. yes p is a prime.

and the order is because i have possibilities for the two elements on the diagonal, and possibible values for the element right on the top ...

And i donīt think that your way is unsatisfactory. To be honest... i just didnīt get what you meaned by "showing the finiteness"
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November 4th, 2013, 10:58 AM   #6
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Re: solvable Group of matrices over Z_p

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Originally Posted by Sandra93
oh i forgot this part .. yes p is a prime.
Then indeed N is solvable, by Feit-Thompson theorem.

Quote:
Originally Posted by Sandra93
i just didnīt get what you meaned by "showing the finiteness"
I meant that the derived series is finite. This can be done by calculating commutator subgroups respectively. Try computing G1 = [G, G]. Do you notice any pattern for the elements of G1?
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November 4th, 2013, 11:53 AM   #7
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Re: solvable Group of matrices over Z_p

ok I'll try it:

Let and be in M:

... and now i just try to multiplate this 4 matrices:



=

... where (**) is a giant term , but definitly element of


is this, what you meaned?? and the pattern is that the elements have the same form als the elements in N?
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November 4th, 2013, 12:05 PM   #8
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Re: solvable Group of matrices over Z_p

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is this, what you meaned?? and the pattern is that the elements have the same form als the elements in N?
Not quite. Note that the elements of [M, M] has all 1s in the main diagonal. Now use this fact to show that elements of [[M, M], [M, M]] consists only of the identity element of M. This can be done by taking elements of N, say, X and Y and then calculating [X, Y].

PS : I would think that it is also provable that [M, M] is indeed N, but that'd be harder the other way around.
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