Abstract Algebra Problem (should be easy)?

In summary, the problem is asking to list the elements and their orders in the group GL_n(\mathbb{Z}/2\mathbb{Z}), and show that it is not abelian. GL_n(\mathbb{Z}/2\mathbb{Z}) is the set of invertible n x n matrices with elements in \mathbb{Z}/2\mathbb{Z}. The group G is not mentioned in the problem, it is just a general notation for a group. The task is to find the orders of the elements in GL_n(\mathbb{Z}/2\mathbb{Z}) and prove that it is not abelian.
  • #1
DEMJ
44
0

Homework Statement



List all the elements of [tex]GL_N(\mathbb{Z}/2\mathbb{Z})[/tex]. Find the order of each element, and show it is not abelian.


The Attempt at a Solution



I am confused right from the get go about [tex]GL_n(\mathbb{Z}/2\mathbb{Z})[/tex].

I think the [tex]L_n(\mathbb{Z}/2\mathbb{Z})[/tex] part means there are a n x n matrices whose elements are [tex]\mathbb{Z}/2\mathbb{Z}[/tex]. Is that correct to say? Also what does the group [tex]G[/tex] have to do in the problem? Any help is appreciated because I am struggling atm to even get started on this problem.
 
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  • #2
I don't know much about finite fields but GLn(F) is the "General Linear" Group and is the set of invertible (i.e. non-singular) matrices with components in F.
 

Related to Abstract Algebra Problem (should be easy)?

1. What is abstract algebra?

Abstract algebra is a branch of mathematics that studies algebraic structures such as groups, rings, and fields. It focuses on the properties and operations of these structures, rather than specific numbers or equations.

2. What are the main topics in abstract algebra?

The main topics in abstract algebra include group theory, ring theory, field theory, and linear algebra. Other topics that are often studied include group actions, Galois theory, and homological algebra.

3. What is the difference between abstract and concrete algebra?

Abstract algebra deals with abstract structures and their properties, while concrete algebra deals with specific numbers and equations. Abstract algebra is more general and can be applied to a wide range of problems, while concrete algebra is more specific and focuses on solving equations.

4. What are some applications of abstract algebra?

Abstract algebra has many applications in mathematics, physics, computer science, and cryptography. It is used to study symmetry in geometry, to understand the structure of molecules, and to develop encryption algorithms.

5. Can you provide an example of an abstract algebra problem?

An example of an abstract algebra problem could be finding the inverse of a matrix in a field, or proving the existence of a group homomorphism between two groups. These problems involve using the properties and operations of abstract algebraic structures to solve for unknown elements or to prove mathematical statements.

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