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

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SUMMARY

The discussion centers on identifying automorphisms of order 2 in the group GL_3(F_2). It clarifies that the inverse automorphism is not an automorphism but an anti-automorphism. Additionally, the conversation touches on the uniqueness of automorphisms of order 2 in GL_2(F_3) and the classification of groups of order 147, specifically regarding the structure of the Sylow subgroups and their automorphisms.

PREREQUISITES
  • Understanding of group theory concepts, particularly automorphisms and anti-automorphisms.
  • Familiarity with the general linear group GL_n over finite fields.
  • Knowledge of Sylow theorems and their application in group classification.
  • Basic understanding of isomorphism and group structure.
NEXT STEPS
  • Research the properties of automorphisms in GL_n(F_q) for various n and q.
  • Study the classification of groups of order 147 and their Sylow subgroups.
  • Learn about the structure and properties of elementary abelian groups.
  • Explore the concept of semidirect products in group theory.
USEFUL FOR

Mathematicians, particularly those specializing in group theory, algebraists, and students studying finite groups and their automorphisms.

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Can anybody help me to find the automorphisms of order 2 in Gl_3 (F_2)? Is it the inverse automorphism?
 
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I thinmk you mean Auts *of* GL. X-->X^-1 is not an automorphism, it is an anti-automorphism.
 
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.
 
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