Group of Automorphisms of an Infinite Cyclic Group - Chapter 10 - Lecture 4

In this lecture we determine the group of Automorphisms of an infinite cyclic group G. We prove that there are precisely two automorphisms of G viz. the identity map and the function which takes every element to its inverse. Thus the group of Automorphisms of G is group of order 2 which is a prime number. We use the fact that every group of prime order is cyclic. Hence the group of automorphisms of G is a cyclic group of order 2. Finally we use the result that every finite cyclic group of order n is isomorphic to Zn the group of integers modulo n w.r.t. addition modulo n. Thus the group of automorphisms of G is isomorphic to (Z_2, +_2).

Link to the result that if a is a generator of a group G then so is its inverse:
Link to proof of every group of prime order is cyclic:

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Link to the first lecture of this chapter