Is the Solution for Problem 2 on the Algebra Qualifier Exam Correct?

In summary, a solvable group of matrices is a group of matrices that can be transformed into an upper triangular form by applying elementary row operations. Solvability in a group of matrices is determined by the existence of a chain of subgroups, where each subgroup is normal and the quotient group of each pair is abelian. These groups have applications in mathematics and physics, but not all groups of matrices are solvable. The concept of solvable groups is also related to the solvability of polynomial equations through the use of Galois groups.
  • #1
mrbohn1
97
0
I'm looking at solutions to an algebra qualifying exam someone has posted on the web; the page is here:

http://mathwiki.gc.cuny.edu/index.php/Spring_2007_Algebra_Qualifier"

I'm looking at problem 2.

Is this solution OK? The author has not addressed the necessary condition that the solvable series has cyclic quotient groups, and I'm not sure that the series he has constructed does.
 
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  • #2
Try to prove this: Let G be a finite group. If G has a solvable series where the quotients are all abelian, then G has a solvable series where the quotients are all cyclic.
 

1. What is a solvable group of matrices?

A solvable group of matrices is a group of matrices that can be transformed into an upper triangular form by repeatedly applying elementary row operations. This means that all the elements in the group can be expressed as a combination of upper triangular matrices with 1s on the main diagonal.

2. How is solvability determined in a group of matrices?

Solvability in a group of matrices is determined by the existence of a chain of subgroups, where each subgroup is a normal subgroup of the previous one and the quotient group of each pair of subgroups is abelian. Once this chain reaches the trivial subgroup, the group is considered solvable.

3. What are the applications of solvable groups of matrices?

Solvable groups of matrices have applications in various areas of mathematics, including group theory, linear algebra, and representation theory. They are also used in physics, particularly in the study of quantum mechanics and symmetry groups.

4. Can all groups of matrices be made solvable?

No, not all groups of matrices are solvable. In fact, there are groups of matrices that are not solvable, such as the special linear group and the general linear group. Solvability is a special property that only some groups of matrices possess.

5. How does the concept of solvable group of matrices relate to the solvability of polynomial equations?

The concept of solvable group of matrices is closely related to the solvability of polynomial equations. This is because a polynomial equation can be solved by finding a group of matrices with certain properties, known as a Galois group. If this group is solvable, then the polynomial equation is considered solvable by radicals.

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