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Problem: Find all elements of the group \( S_{n} \), permutable with the cycle \( \left(\alpha_{1} \alpha_{2} \ldots \alpha_{n}\right) \), where \( \left(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}\right) \) - the permutation of numbers \( 1,2, \ldots, n \).

1.3.12 Permutation group

3.75 $

Problem: Prove that if the group \( G \) is non-commutative and \( |G|=8 \), then \( G \cong D_{4} \) or \( G \cong Q_{8} \).

1.3.13 Permutation group

4.49 $

Problem: Prove that Aut \( S_{3}=\operatorname{Inn} S_{3} \cong S_{3} \).

1.3.14 Permutation group

2 $

Problem: Prove that two permutations are conjugate in a group \( S_{n} \) if and only if they have the same cycle structure (i.e. their decomposition into products of independent cycles for any \( k \) contains the same number of cycles of length \( k \) ).

1.3.15 Permutation group

2.5 $

Problem: Find rational roots of a polynomial: \[ P(x)=-9 x^{4}+6 x^{3}-23 x^{2}+4 x+4 . \]

1.10.1 Polynomials

1.5 $

Problem: Find rational roots of a polynomial: \[ P(x)=2 x^{4}-5 x^{3}+4 x^{2}+3 x+9 . \]

1.10.2 Polynomials

1.5 $

Problem: Find GCD of polynomials \( f(x), g(x) \in \mathrm{F}_{2}[x] \), where \[ \begin{array}{l} f(x)=x^{7}+x^{5}+x^{4}+x+1, \\ g(x)=x^{8}+x^{6}+x^{5}+x^{4}+x^{3}+x^{2}+1 . \end{array} \]

1.10.3 Polynomials

2.5 $

Problem: The action of a stationary linear system is given as follows: \( \xi^{\prime \prime}+2 \xi^{\prime}+2 \xi=\eta^{\prime}+\eta \). The correlation function of the output is known: \( K_{\eta, \eta}(\tau)=\exp (-|\tau|), \tau \in(-\infty ;+\infty) \). Find the correlation function of the input.

15.1.24 Theory of random processes

6.24 $

Problem: Find the spectral density of the random function \( \xi(t) \), if \( K_{\eta, \eta}(\tau)=\exp (-2|\tau|) \). \[ \xi^{\prime \prime}+\xi=\eta^{\prime}+2 \eta \text {. } \]

15.1.25 Theory of random processes

3 $

Problem: \( \xi^{I V}+4 \xi=\eta^{\prime}+2 \eta \). The random function of the output is \( \eta(t) \). Find the spectral density of the input \( S_{\xi}(\omega) \), if \( K_{\eta, \eta}(\tau)=e^{-2|\tau|} \).

15.1.26 Theory of random processes

3.75 $

Problem: A spectral density of the random function \( \xi(t) \) is given: \[ S_{\xi}(\omega)=\frac{1}{\pi\left(4+\omega^{4}\right)} . \] Find the correlation function of this random function.

15.1.27 Theory of random processes

4.99 $

Problem: Vector \( X=(7 ; 7 ; 2) \) is given in the basis \( e=\left\{e_{1}, e_{2}, e_{3}\right\} \). Find the coordinates of \( X \) in the basis of \( e^{\prime}=\left\{e_{1}^{\prime}, e_{2}^{\prime}, e_{3}^{\prime}\right\} \), where \[ \left\{\begin{array}{l} e_{1}^{\prime}=e_{1}+e_{2}+\frac{6}{7} e_{3} \\ e_{2}^{\prime}=-6 e_{1}-e_{2} \\ e_{3}^{\prime}=-e_{1}+e_{2}+e_{3} \end{array}\right. \]

1.1.5 Vector Algebra

1 $

Problem: Find out whether a group is formed by the following set under the specified operation on elements: \{Integer numbers that are multiples of the given natural number \( n \), with respect to the additions\}.

1.6.19 Fields, Groups, Rings

1.25 $

Problem: Find the centre of the groups: a) \( 0(2) \) is a group of orthogonal matrices \( 2 \times 2 \), b) \( S U(2)=\left\{A \in G L_{2}(\mathbb{R})|| A \mid=1\right\} \).

1.6.20 Fields, Groups, Rings

6.24 $

Problem: Find all such finite groups \( G \) that the number of conjugacy classes in \( G \) is equal to 1,2 and 3 .

1.6.21 Fields, Groups, Rings

3.25 $

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