Section 1: Algebra

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Geometry || Analysis

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1. Which of the following statements are true?
   1. Every group of order 11 is cyclic.
   2. Every group of order 111 is cyclic.
   3. Every group of order 1111 is cyclic.
      

      Discussion

2. Let $latex S_n $  denote the symmetric group of order n, i.e. the group of all permutations of the n symbols (1, 2, ...  , n). Given two permutations $latex sigma $ and $latex tau $ in $latex S_n $, we define the product $latex sigma tau $ as their composition got by applying $latex sigma $ first and then applying $latex tau $ to the set {1,2, ... , n}, Write down the following permutation in $latex S_8 $ as the product of disjoint cycles:
   (1 4 3 8 7)(5 4 8).
3. Write down all the permutations in $latex S_4 $ which are conjugate to the permutation (1 2)(3 4).
4. Let R be a ring such that $latex x^2 = x $ for every $latex x \in R $. Which of the following statements are true?
   1. $latex x^n $ for every $latex n \in N $ and every $latex x \in R $
   2. x= -x for every $latex x \in R $
   3. R is a commutative ring.
5. For a prime number p let $latex F_p $ denote the field consisting of 0, 1, 2, ... ,  p - 1 with addition and multiplication modulo p. Which of the following quotient rings are fields?
   1. $latex F_5 [x] / (x^2 + x + 1) $
   2. $latex F_2 [x] / (x^3 + x + 1) $
   3. $latex F_3 [x] / (x^3 + x + 1) $
6. Let V be the subspace of $latex M_2 (R) $ consisting of all matrices with trace zero and such that all entries of the first row add up to zero. Write down a basis for V.
7. Let  V subset of $latex M_n (R) $ be a subspace of all matrices such that the entries in every row add up to zero and the entries in every column also add up to 0. What is the dimension of V?
8. Let T : $latex M_2 (R) $ --> $latex M_2 (R) $ be a linear transformation defined by $latex T(A) = 2A + 3A^T $ . Write down the matrix of this transformation with respect to the basis { $latex E_i , 1 \le i \ge 4 $ } where $latex E_1 $ = $latex \begin {bmatrix} 1 & 0 0 & 0 \end {bmatrix} $ , $latex E_2 $ = $latex \begin {bmatrix} 0 & 1 0 & 0 \end {bmatrix} $ , $latex E_3 $ = $latex \begin {bmatrix} 0 & 0 1 & 0 \end {bmatrix} $ , $latex E_4 $ = $latex \begin {bmatrix} 0 & 0 1 & 0 \end {bmatrix} $
9. Find the values of $latex \alpha\in R $ such that the matrix $latex \begin {bmatrix} 3 & \alpha \alpha & 5 \end {bmatrix} $ has 2 as an eigenvalue.
10. Let $latex A = (a_{ij} ) \in M_3 (R) $ be such that $latex a_{ij} = - a_{ji} $ for all $latex 1 \le i , j \le 3 $. If 31 is a eigenvalue of A find it's other eigenvalues.

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Geometry || Analysis

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