The image shows a question in Russian that translates to: "Indicate all correct statements," followed by four mathematical statements related to permutations. Here's a breakdown of the statements and their correctness:

1. Любая подстановка раскладывается в произведение независимых циклов единственным образом
   Translation: "Any permutation can be decomposed into a product of independent cycles in a unique way."
   Correct: This is a fundamental property of permutations. Any permutation can be uniquely decomposed into disjoint cycles.

2. $\varepsilon(\sigma) = -\varepsilon(\sigma^{-1})$
   Translation: "$\varepsilon(\sigma) = -\varepsilon(\sigma^{-1})$"
   Correct: This is true because the sign of a permutation ($\varepsilon$) changes for the inverse of the permutation.

3. Пусть дана подстановка $\sigma$ и транспозиция $\tau$, тогда $\varepsilon(\sigma\tau) = -\varepsilon(\sigma)$
   Translation: "Let $\sigma$ be a permutation and $\tau$ a transposition, then $\varepsilon(\sigma\tau) = -\varepsilon(\sigma)$"
   Correct: This is true. Multiplying a permutation by a transposition changes its sign because transpositions are odd permutations.

4. Подстановка — это биективное преобразование
   Translation: "A permutation is a bijective transformation."
   Correct: This is true. By definition, a permutation is a bijection, meaning it is both injective and surjective.

Conclusion:

All four statements are correct.

Do you have any further questions or need more details on any of the concepts?

Here are 5 related questions:

1. What is the cycle decomposition of a permutation?
2. How does the sign of a permutation relate to its cycle structure?
3. What is the formula for the number of transpositions in a permutation?
4. Can every permutation be written as a product of transpositions?
5. How does the inverse of a permutation affect its decomposition into cycles?

Tip: Remember that a transposition (a swap of two elements) is always an odd permutation!