From the uploaded image, the task is to identify correct mathematical statements about groups and group properties. Let’s analyze each statement:

1. "Множество квадратных матриц второго порядка с определителем равным 1 является группой."

   • The set of $2 \times 2$ matrices with determinant equal to 1 forms a group under matrix multiplication. This is true because:
     • Matrix multiplication is associative.
     • The determinant condition ensures the existence of an inverse matrix within the set.
     • The identity matrix (determinant = 1) is included in the set.
   • This statement is TRUE.

2. "Множество квадратных матриц вида $\begin{pmatrix} a & b \\ 0 & c \end{pmatrix}$, $a, b, c \in \mathbb{Z}$" образуют группу относительно умножения матриц.

   • To verify if this forms a group under multiplication:
     • Closure: Multiplying two such matrices results in a matrix of the same form.
     • Associativity: Matrix multiplication is associative.
     • Identity: The identity matrix ($\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$) is not guaranteed unless specific constraints on $a, b, c$ exist.
     • Inverses: Not all matrices of this form have inverses in the integer domain $\mathbb{Z}$.
   • This statement is FALSE.

3. "Множество действительных чисел на отрезке [-1, 1] с введенной операцией сложения является абелевой группой."

   • For the interval $[-1, 1]$ under addition:
     • Closure: Adding two numbers from $[-1, 1]$ may exceed the interval (e.g., $0.9 + 0.5 = 1.4$).
     • Other group axioms fail due to the failure of closure.
   • This statement is FALSE.

4. "Множество делителей числа 6 с введенной операцией умножения чисел является группой."

   • The divisors of 6 are $\{1, 2, 3, 6\}$.
   • Closure: Multiplying two divisors results in another divisor of 6.
   • Associativity: Multiplication is associative.
   • Identity: The identity (1) exists.
   • Inverses: Not all divisors have inverses (e.g., $2 \times 3 = 6$, but no inverse exists in this set for multiplication).
   • This statement is FALSE.

Summary of Correct Statements:

• 1. TRUE
• 2, 3, 4. FALSE

Would you like me to explain any specific step in more detail or provide further examples?

Related Questions:

1. What are the conditions for a set to be considered a group?
2. How does the determinant property relate to the invertibility of a matrix?
3. Can you provide examples of other mathematical objects that form Abelian groups?
4. Why does closure fail for the addition operation in bounded intervals like $[-1, 1]$?
5. What are the key differences between Abelian and non-Abelian groups?

Tip:

When analyzing group properties, always start by checking closure and the existence of an identity, as these are often the quickest ways to determine if the set fails to be a group.