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

tumpabhat

Can anybody help me to find the automorphisms of order 2 in Gl_3 (F_2)? Is it the inverse automorphism?

matt grime

I thinmk you mean Auts *of* GL. X-->X^-1 is not an automorphism, it is an anti-automorphism.

tumpabhat

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 donot 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 donot know latex. So, I will appreciate if you kindly post your reply in just word.