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

I have come up with the Question no. 2 of four problems related to Sylow’s theorem with high difficulty level. Let’s take a look at the problem and understand the concept. Let P be a Sylow p- group of a finite group G and let H be a subgroup of G...
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 6

Arithmetical Dynamics: Part 6

Arithmetical dynamics is the combination of dynamical systems and number theory in mathematics. Again, we are here with the Part 6 of the Arithmetical Dynamics Series. Let’s get started…. Consider fix point of \( R(z) = z^2 – z \) . Which is the...