How Might I Show That Aut(Z) Has Order 2?

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Homework Help Overview

The discussion revolves around proving that the automorphism group of the integers, Aut(Z), has order 2. Participants are exploring the properties and mappings that define this group.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the generators of Z and the potential mappings in Aut(Z), specifically considering the functions f(x) = x and f(x) = -x. There are questions about how to formally demonstrate that these mappings constitute all elements of Aut(Z).

Discussion Status

The conversation includes attempts to clarify the nature of the elements in Aut(Z) and whether the provided mappings sufficiently demonstrate the group's order. Some participants assert that the two identified mappings imply the order of the group, while others seek further justification for this conclusion.

Contextual Notes

There is an implicit assumption that the properties of automorphisms and generators are understood, but no specific equations or formal definitions are provided in the posts.

PennState666
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Homework Statement


prove Aut(Z) has order 2.


Homework Equations



none

The Attempt at a Solution


The generators for Z = <-1, 1>.
if f is a mapping in Aut(Z), f(x)= x or f(x) = -x
 
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Sure, the two elements of Aut(Z) are f(x)=x, and f(x)=(-x). Any question?
 
how might I show that Aut(Z) has order 2?
 
PennState666 said:
how might I show that Aut(Z) has order 2?

You just did. Doesn't that show that there are two elements in Aut(Z)? A generator must map to a generator if you are going to get an automorphism.
 

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