4 questions from Sylow’s theorem: Qn 4

4 questions from Sylow’s theorem: Qn 4

Prove that if |G| = 8000 then G is not simple . SOLUTION If \( |G| = 2^3 \times 10^3 = 2^6 \times 5^3 \\ consider \ , \ n_5 = (5k+1) | 2^6 \\ n_5 = 1 , 16 \\ n_2 = (2k +1) | |G| \\ \Rightarrow n_2 = 5 \ , 25 ,\ 125 \) . Let , H and K are two Sylow – 5- subgroups...
4 questions from Sylow’s theorem: Qn 3

4 questions from Sylow’s theorem: Qn 3

Prove that if |G| = 2376 then G is not simple . SOLUTION \( |G| = 2376 = 2^3 \times 3^3 \times 11 \) If \( n_{11} = 12 \\ \\ Let \ , H \in Syl_{11}(G) \ then \ consider \ \ N_G (H) ; [ G : N_G(H) ] \\ n_{11} = 12 \\ \Rightarrow | N_G(H) | = \frac {2376}{12} = 198 \\...
4 questions from Sylow’s theorem: Qn 2

4 questions from Sylow’s theorem: Qn 2

Let P be a Sylow p- group of a finite group G and let H be a subgroup of G containing \( N_{G}(P) \) . Prove that \( H = N_{G}(H) \). Solution Let \( P \in Syl_{P}(G) \ and H \leq G \ such \ that \ N_{G}(P) \subset H \) Claim : Frattinis Argument : If G is a finite...
4 questions from Sylow’s theorem: Qn 1

4 questions from Sylow’s theorem: Qn 1

Prove that if |G| = 616 then G is not simple . SOLUTION \( |G| = 616 = 2^3 \times 7 \times 11 \) Consider the 11 – sylow subgroup of G . \( n_{11} \mid |G| = 616 \ \ n_{11} = number \ of \ sylow_ {11} subgroup \\ \Rightarrow (11k + 1) \mid 56 \ \Rightarrow n \in...
Arithmetical Dynamics: Part 0

Arithmetical Dynamics: Part 0

Rational function \( R(z)= \frac {P(z)}{Q(z)} \) ; where P and Q are polynimials . There are some theory about fixed points . Theorem: Let \( \rho \) be the fixed point of the maps R and g be the Mobius map . Then \( gRg^{-1} \) has the same number of fixed points at...