The task is to identify which statements are correct based on the group theory concepts presented in the image. Let’s analyze each statement in turn.

1. First Statement:

   • The statement concerns the group of positive rational numbers under multiplication with a mapping $f(x) = x^{-1}$.
   • Since $f(x) = x^{-1}$ (the reciprocal function) is bijective and preserves the operation in this group (as $(x \cdot y)^{-1} = x^{-1} \cdot y^{-1}$), it is indeed an automorphism in this context.
   • This statement is correct.

2. Second Statement:

   • This statement addresses the composition of automorphisms, claiming that the composition of any number of automorphisms is also an automorphism.
   • In group theory, the set of automorphisms of a group forms a group itself under composition, meaning the composition of automorphisms is indeed an automorphism.
   • This statement is correct.

3. Third Statement:

   • This statement refers to the group of 2x2 matrices with matrix addition and defines a mapping $f(A) = A^T$ (transpose of $A$).
   • In the context of matrix addition, taking the transpose of a matrix does not affect the addition operation since addition is commutative. Thus, $f(A) = A^T$ is indeed an automorphism in the group of matrices under addition.
   • This statement is correct.

4. Fourth Statement:

   • This statement pertains to automorphisms of cyclic groups, suggesting that under an automorphism, generating elements can map to the neutral element.
   • This is incorrect because an automorphism of a cyclic group maps generators to other generators, not to the neutral element, as the structure of the group must be preserved.
   • This statement is incorrect.

Summary of Correct Statements

The correct statements are:

• The first statement
• The second statement
• The third statement

Would you like a more detailed explanation for any of these statements or to explore related concepts?

Here are five follow-up questions to expand your understanding:

1. Why does the reciprocal function act as an automorphism in the group of positive rational numbers under multiplication?
2. What properties make the composition of automorphisms result in another automorphism?
3. How does the transpose operation affect matrix properties under different operations?
4. Why must an automorphism in a cyclic group map generators to generators?
5. What are some examples of other groups where automorphisms are commonly studied?

Tip: In group theory, always verify if a mapping preserves the group operation to confirm if it’s an automorphism.