Solution

The correct answer is 2

Key PointsIn the group of all integers under addition, there are only two automorphisms. One is the identity map (each number maps to itself) and the other is the negation map (each number maps to its negative). This is because an automorphism in this group must preserve the group operation, which is addition, and these two functions are the only ones that do this. 

EXPLANATION:

• An automorphism is an isomorphism from a mathematical structure onto itself. In the context of groups, it's an isomorphism from a group to itself. In terms of the algebraic structure, an isomorphism preserves the group operation. For the group of integers under addition (Z, +), the operation is addition.
• In case of (Z, +), an automorphism must map 0 to 0 (as every group homomorphism preserves the identity) and must preserve addition. There are only two homomorphisms from (Z, +) to itself:
  • The identity mapping (the function that maps every integer to itself), f(n) = n, and
  • The negation function, f(n) = -n.
• This is because if τ: ℤ→ℤ is a group homomorphism, then τ(n) = nτ(1) for all n ∈ ℤ. As a result, the value of τ on any integer is determined by the image of 1.

Both of these homomorphisms are in fact automorphisms since they are bijective (one-to-one and onto).

So, the correct answer is 2) 2.