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November 4th, 2013, 02:03 AM  #1 
Newbie Joined: Nov 2013 Posts: 26 Thanks: 0  solvable Group of matrices over Z_p
show that the set of invertible upper triangular matrices over is a Group for matrixmultiplication.\\ so I showed that the set is a group by showing the groupaxioms. 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 subgroupcriterium+ 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] 
November 4th, 2013, 04:19 AM  #2 
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: solvable Group of matrices over Z_p
Actually, a general result holds true : The group of invertible uppertriangular martices over any finite field is solvable. Try showing the finiteness of the derived series of M. 
November 4th, 2013, 04:58 AM  #3 
Newbie Joined: Nov 2013 Posts: 26 Thanks: 0  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 .. 
November 4th, 2013, 06:14 AM  #4  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: solvable Group of matrices over Z_p Quote:
Quote:
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November 4th, 2013, 09:43 AM  #5 
Newbie Joined: Nov 2013 Posts: 26 Thanks: 0  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" 
November 4th, 2013, 10:58 AM  #6  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: solvable Group of matrices over Z_p Quote:
Quote:
 
November 4th, 2013, 11:53 AM  #7 
Newbie Joined: Nov 2013 Posts: 26 Thanks: 0  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? 
November 4th, 2013, 12:05 PM  #8  
Math Team Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory  Re: solvable Group of matrices over Z_p Quote:
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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