Automorphisms of (Z, +)

An automorphism of a group is an isomorphism from the group to itself, i.e., a bijective map that preserves the group structure. In this case, we are considering the group (Z, +), where Z represents the set of all integers and + denotes addition.

Properties of (Z, +)

1. Closure: For any two integers a and b, the sum a + b is also an integer.
2. Associativity: For any three integers a, b, and c, (a + b) + c = a + (b + c).
3. Identity element: The integer 0 is the identity element, such that for any integer a, a + 0 = 0 + a = a.
4. Inverse element: For any integer a, there exists an integer -a such that a + (-a) = (-a) + a = 0.

Structure of (Z, +)

The group (Z, +) is an infinite cyclic group generated by the integer 1. This means that every integer can be expressed as a sum of 1's or their inverses. For example, 5 = 1 + 1 + 1 + 1 + 1, and -3 = -1 - 1 - 1.

Number of Automorphisms

1. Identity Automorphism: The identity automorphism is the map that sends each integer to itself. It preserves the group structure and is always an automorphism. Therefore, there is at least one automorphism.

2. Negation Automorphism: The negation automorphism is the map that sends each integer a to its negation -a. This map is bijective, preserves the group structure, and is an isomorphism. Therefore, there is at least one more automorphism.

3. No Other Automorphisms: Any other map from (Z, +) to itself must preserve the group structure, meaning it should preserve the sum of two integers. However, there are no other bijections that satisfy this condition. If a map sends 1 to a, it must also send n (the sum of n 1's) to na for any integer n. However, there are no other integers that can satisfy this condition for all integers. Therefore, there are no other automorphisms.

Conclusion

In conclusion, the number of automorphisms of the group (Z, +) is 2, namely the identity automorphism and the negation automorphism. Option B is the correct answer.