An excursion in Linear Algebra

An excursion in Linear Algebra

Pick up any odd book on linear algebra. Matrices, base changes, eigen-values will pop up. It is hard to appreciate why an intelligent person should spend hours and days, mastering these skills. Our excursion, begins with an attempt to understand that very thing:...
Linear Algebra total recall (Open Seminar)

Linear Algebra total recall (Open Seminar)

14th November 2019 8 PM. I.S.T. Online, Live Seminar Understand We will review major ideas from linear algebra with easy examples. It is part of our College Mathematics Program seminar sequence. Who may attend? There are 50 seats. The seminar itself is free and open...
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...