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# Section 1: Algebra

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