Orthogonal group over finite field

In summary, the orthogonal group over a finite field is a mathematical concept used to describe the symmetries of a finite-dimensional vector space over a finite field. It consists of all invertible linear transformations that preserve the length of vectors in the space. The size of the orthogonal group over a finite field is determined by the order of the field and the dimension of the vector space. This group has many applications in areas such as coding theory, cryptography, and combinatorics.
  • #1
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Let O(n,F_q) be the orthogonal group over finite field F_q. The question is how to calculate the order of the group.
The answer is given in http://en.wikipedia.org/wiki/Orthogonal_group#Over_finite_fields". This seems to be a standard result, but I could not find a proof for this in the basic representation theory books that I have. Neither could I solve it myself from the (direct sum) construction they have given.
Can someone please help me? It is enough if you give some references (books/papers) where it is solved.
 
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  • #2
It is not within textbooks as it is comparably uninteresting. I recommend an induction over the dimension to prove it.
 

1. What is an orthogonal group over a finite field?

The orthogonal group over a finite field is a mathematical concept that describes the set of all linear transformations that preserve the length of vectors in a vector space over a finite field. In other words, it is the group of all matrices that represent rotations, reflections, and other types of transformations that do not change the size or shape of a vector.

2. How is the orthogonal group over a finite field different from the orthogonal group over a real or complex field?

The main difference between the orthogonal group over a finite field and over a real or complex field is that the finite field contains a limited number of elements, whereas real and complex fields are infinite. This means that the elements of the finite field are discrete, while the elements of real and complex fields are continuous. This has implications for the properties and applications of the orthogonal group in different fields of mathematics and science.

3. What are the applications of the orthogonal group over a finite field?

The orthogonal group over a finite field has various applications in different fields of mathematics and science. It is used in coding theory and cryptography for constructing error-correcting codes and secure communication protocols. It also has applications in combinatorics, number theory, and algebraic geometry. In physics, it is used to study the symmetries of physical systems and in quantum computing for implementing quantum gates.

4. What are the properties of the orthogonal group over a finite field?

The orthogonal group over a finite field has several important properties that make it a useful mathematical concept. It is a finite group, meaning that it has a finite number of elements. It is also a non-abelian group, meaning that the order in which operations are performed matters. Additionally, it has a subgroup structure that can be described using the theory of vector spaces over finite fields.

5. How is the orthogonal group over a finite field related to other mathematical concepts?

The orthogonal group over a finite field is closely related to other mathematical concepts such as finite fields, vector spaces, and group theory. It can be represented using matrices, which are elements of the general linear group over the same finite field. It also has connections to other groups such as the special orthogonal group and the symplectic group. Overall, the orthogonal group over a finite field plays an important role in understanding and solving problems in various branches of mathematics.

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