Abstract Algebra Problem (should be easy)?

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SUMMARY

The discussion centers on the group GL_N(ℤ/2ℤ), which consists of all n x n invertible matrices over the finite field ℤ/2ℤ. Participants clarify that GL_N(ℤ/2ℤ) includes matrices with elements from the field ℤ/2ℤ, and they emphasize the importance of understanding the group's structure, particularly its non-abelian nature. The order of each element in this group is also a focal point, as it is crucial for demonstrating the group's properties.

PREREQUISITES
  • Understanding of finite fields, specifically ℤ/2ℤ.
  • Knowledge of group theory, particularly the properties of general linear groups.
  • Familiarity with matrix operations and invertibility.
  • Basic concepts of non-abelian groups.
NEXT STEPS
  • Study the structure and properties of GL_N(ℤ/2ℤ) in detail.
  • Learn how to compute the order of elements in finite groups.
  • Explore examples of non-abelian groups and their characteristics.
  • Investigate the relationship between matrix groups and linear transformations.
USEFUL FOR

Students of abstract algebra, mathematicians interested in group theory, and anyone studying linear algebra with a focus on finite fields and matrix groups.

DEMJ
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Homework Statement



List all the elements of GL_N(\mathbb{Z}/2\mathbb{Z}). 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 GL_n(\mathbb{Z}/2\mathbb{Z}).

I think the L_n(\mathbb{Z}/2\mathbb{Z}) part means there are a n x n matrices whose elements are \mathbb{Z}/2\mathbb{Z}. Is that correct to say? Also what does the group G 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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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.
 

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