Please help me find the automorphisms of order 2 in Gl_3 (F_2)

  • Thread starter tumpabhat
  • Start date
In summary, the conversation discusses finding the automorphisms of order 2 in Gl_3 (F_2) and whether they are unique in Gl_2(F_3). The topic also touches on classifying groups of order 147 and the use of homomorphisms in finding automorphisms. The user also requests for the reply to be in plain text instead of latex format.
  • #1
tumpabhat
2
0
Can anybody help me to find the automorphisms of order 2 in Gl_3 (F_2)? Is it the inverse automorphism?
 
Physics news on Phys.org
  • #2
I thinmk you mean Auts *of* GL. X-->X^-1 is not an automorphism, it is an anti-automorphism.
 
  • #3
group classification

well, thank you so much. Yup, now I realize that.
Also I have a question. Are the automorphisms of order 2 unique in Gl_2(F_3) ? Are two elements of order 2 conjugate in Gl_2_(F_3).
I don't realize that...
I was trying to calssify all groups of order 147 upto isomorphism. Now, if P denotes the unique normal Sylow 7 subgrp of order 49 then P is either ismorphic to C_49 or is the elementary abelian group Z_7 times Z_7. Now if Q is sylow 3 subgrp of order 3 and T :Q to Aut(P) denote the homomorphism then fro any q in Q order of T(q) is either 1 ( we have trivial homomorphism and that gives G iso C_49 times C_3 = C_147.)
But if order of T(q) = 3 in Aut (P) then how do I find an automorphism of order 3 in
Aut(P) and how do I know where that automorphism sends p to? Because if P=<p>, and Aut(P)= <Y> then Y(p)= ?
I want to write G iso C_49 semidirect C_3 but for that I domot know the generators and relation.
Also I don't know latex. So, I will appreciate if you kindly post your reply in just word.
 
Last edited:

1. What is an automorphism?

An automorphism is a mathematical concept that refers to a function or mapping that preserves the structure of a mathematical object. In other words, an automorphism is a transformation that maintains the properties and relationships of a given object.

2. What is the order of an automorphism?

The order of an automorphism refers to the number of times the automorphism can be applied to a mathematical object before returning to its original state. In this case, we are looking for automorphisms of order 2, meaning they can be applied twice before returning to the original object.

3. What is Gl3(F2)?

Gl3(F2) refers to the general linear group of 3x3 matrices over the field F2, which consists of all invertible matrices with entries in the finite field F2 (also known as the Galois field with 2 elements).

4. How do you find automorphisms of order 2 in Gl3(F2)?

To find automorphisms of order 2 in Gl3(F2), we can use the fact that any automorphism can be represented as a matrix with respect to a chosen basis. Therefore, we can look for matrices in Gl3(F2) that, when squared, will result in the identity matrix.

5. What are some examples of automorphisms of order 2 in Gl3(F2)?

Some examples of automorphisms of order 2 in Gl3(F2) include the matrix [1 0 0; 0 1 0; 0 0 1], which represents the identity automorphism, and the matrix [1 1 0; 1 0 0; 1 1 1], which represents a reflection across the line x=y. There are many more possible automorphisms of order 2 in Gl3(F2), but these are just a few examples.

Similar threads

Replies
2
Views
844
  • Linear and Abstract Algebra
Replies
2
Views
1K
  • Linear and Abstract Algebra
Replies
5
Views
2K
  • Linear and Abstract Algebra
Replies
5
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
2K
  • Linear and Abstract Algebra
Replies
19
Views
3K
  • Linear and Abstract Algebra
Replies
2
Views
2K
Replies
3
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
2
Views
1K
Back
Top