Solubility of the General Linear Group

Thread starter herbert_454
Start date Nov 23, 2008
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General Group Linear Solubility

In summary, the General Linear Group (GL(n)) is a mathematical group consisting of invertible n by n matrices over a specified field. The solubility of GL(n) is significant in various fields such as group theory, representation theory, and quantum mechanics, and has applications in computer science and engineering. The solubility of GL(n) is determined by the solvability of its subgroup, the Special Linear Group (SL(n)), and a solvable group can be broken down into simpler subgroups through a solvable series. Examples of solvable GL(n) include those over a finite field and the Special Linear Group over any field, while those over the real or complex numbers are non-solvable.

Nov 23, 2008

herbert_454

I am interested, in particular, why GL_2(3) (or in other notation GL(2,3)) is soluble. Can anyone explain why or recommend a textbook with a pretty proof. Thank you sincerely.

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Nov 23, 2008

morphism

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This follows from http://planetmath.org/encyclopedia/BurnsidePQTheorem.html .

Last edited by a moderator: May 3, 2017

Nov 24, 2008

herbert_454

thanks!

Related to Solubility of the General Linear Group

1. What is the General Linear Group?

The General Linear Group, denoted as GL(n), is a mathematical group that consists of all invertible n by n matrices over a specified field.

2. What is the significance of the solubility of the General Linear Group?

The solubility of the General Linear Group is important in various areas of mathematics and physics, such as group theory, representation theory, and quantum mechanics. It also has applications in computer science and engineering.

3. How is the solubility of the General Linear Group determined?

The solubility of the General Linear Group is determined by the solvability of its subgroup, the Special Linear Group (SL(n)). If SL(n) is solvable, then GL(n) is also solvable.

4. What is the relationship between solubility and the structure of a group?

Solubility is a property of a group that describes its ability to be "broken down" into simpler subgroups. A solvable group has a chain of subgroups, each normal in the next, that eventually leads to the trivial subgroup. This chain is known as a solvable series.

5. What are some examples of solvable and non-solvable General Linear Groups?

The General Linear Group over a finite field is solvable, as well as the Special Linear Group over any field. On the other hand, the General Linear Group over the real or complex numbers is non-solvable.