Question

Prove that a group of automorphisms of a infinite cyclic group is of order 2.

Answer 1

G isjkite, G/ZG is a non-abelian simple group, and ZG is cyclic. ... Suppose that A is an infinite abelian group and that Aut A is finite. A theorem of Nagrebeckii [ 111 asserts that in an infinite group with finitely many automorphisms the elements of finite order form a finite subgroup.

Answer 2

G isjkite, G/ZG is a non-abelian simple group, and ZG is cyclic. ...

Suppose that A is an infinite abelian group and that Aut A is finite.

A theorem of Nagrebeckii [ 111 asserts that in an infinite group with finitely many automorphisms the elements of finite order form a finite subgroup.

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