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

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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.
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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