Fields, Groups, Rings

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Problem: Are rings \( A \) and \( R \), where \( A \) is a set of complex matrices of the form \( \left(\begin{array}{cc}z & w \\ -\bar{W} & \bar{z}\end{array}\right) \), and \( R \) is a set of real matrices of the form \[ \left(\begin{array}{cccc} x & -y & -z & -t \\ y & x & -t & z \\ z & t & x & -y \\ t & -z & y & x \end{array}\right) ? \]

1.6.18 Fields, Groups, Rings

4.99 $

Problem: Let \( e_{1}, \ldots, e_{n} \) be such elements of the center of the ring \( A \) with such 1 , that \( 1=e_{1}+\cdots+e_{n}, e_{i}^{2}=e_{i} \), \( e_{i} e_{j}=0, i \neq j \). Prove that \( A e_{i} \) are two-sided ideals of the ring \( A \) and \( A=A e_{1} \oplus \ldots \oplus A e_{n} \).

1.6.17 Fields, Groups, Rings

6.24 $

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 $

Problem: Two groups and a mapping between them are given. Is the given mapping a homomorphism, epimorphism, monomorphism, isomorphism? \[ \begin{array}{l} \left(Z_{3},+\right),\left(Z_{9},+\right), \\ f(\overline{0})=\overline{0}, \quad f(\overline{1})=\overline{3}, \quad f(\overline{2})=\overline{6} . \end{array} \]

1.6.22 Fields, Groups, Rings

3 $

Problem: For a given group \( G \) and its subgroup \( H \) : a) describe (left) cosets, b) find out if the factor group is defined, c) if the factor group is defined, give an example of an isomorphic group. \[ G=(G L(n, \mathbb{R}) \cdot \cdot), H=\{A \in G L(n \cdot \mathbb{R}):|\operatorname{det} A|=1\} . \]

1.6.8 Fields, Groups, Rings

4.99 $

Problem: Find out whether the set of integers divisible by 5 , form a group with respect to the addition. \[ M=\{n \in \mathbb{Z} \mid n: 5\}=\{5 k \mid k \in \mathbb{Z}\} . \]

1.6.23 Fields, Groups, Rings

0 $

Problem: Find the ideal (10) in rings \[ (\mathbb{Z} ;+; \cdot) ;(\mathbb{Z}[i] ;+; \cdot) ;(\mathbb{Z}[x] ;+; \cdot) ;(\mathbb{Q}[x] ;+; \cdot) \text {. } \]

1.6.24 Fields, Groups, Rings

2 $

Problem: Prove that the groups \( (\mathbb{Z} ;+) \) and \( (G ; \cdot) \) are isomorphic; where \( G=\left\{x \mid x=2^{t}, t \in \mathbb{Z}\right\} \).

1.6.25 Fields, Groups, Rings

2 $

Problem: Prove that numbers \( 5 ;-5 ; 5 i ;-5 i \) are reducible in the \( \operatorname{ring}(\mathbb{Z}[i] ;+; \cdot) \).

1.6.26 Fields, Groups, Rings

2.5 $

Problem: An algebraic system \( (X, *) \) with one binary operation "*" is given. Indicate what type (groupoid, semigroup, monoid, group) this system belongs to.

1.6.27 Fields, Groups, Rings

2.5 $

Problem: For the given ring \( K \) and its subring \( H \) : a) describe (left) cosets; b) find out whether the quotient ring is defined; c) if a quotient ring is defined, give an example of an isomorphic ring; \( K=\left(\mathbb{R}^{\infty},+, \cdot\right) \) is a ring of all sequences of real numbers, \( \quad H=\left\{\left\{x_{n}\right\} \in \mathbb{R}^{\infty}: x_{1}=\cdots=x_{n}=0\right\} \) ( \( n \) is fixed).

1.6.28 Fields, Groups, Rings

3 $

Problem: Find out whether the set of powers of number 7 with integer exponents form a group with respect to multiplication. \[ M=\left\{7^{n} \mid n \in \mathbb{Z}\right\} \text {. } \]

1.6.29 Fields, Groups, Rings

0 $

Problem: Let \( H \) be a subgroup, generated by the element \( b \) in the multiplicative \( \mathbb{Z}_{p}^{*} \) of residues modulo \( p \), and the \( g H \) coset of the subgroup \( H \), generates by the element \( g \). Calculate the subgroup \( H \) and the coset \( g H \). \[ p=97, b=8, g=2 \text {. } \]

1.6.30 Fields, Groups, Rings

3 $

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