Prove that every automorphism of R*, the group of nonzero real numbers under multiplication, maps positive numbers to positive numbers and negative numbers to negative numbers

Tahmid Knox 2020-10-28 Answered
Prove that every automorphism of R*, the group of nonzero real numbers under multiplication, maps positive numbers to positive numbers and negative numbers to negative numbers
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liannemdh
Answered 2020-10-29 Author has 106 answers

ϕ:RR be an automorphism
Concept used:
An isomorphism from a group G oto itself is called a automorphism of G*
Note that R* is the of nonzero real numbers.
Since ϕ(x)R then obtain ϕ(x)0 for every xR
Let x>0 then obtain x make sense.
Recall the theoram, properties if isomorphism acting on elements. Suppose that ϕ is in isomorphism from a group G ont G.
For every integer n and for every froup of elemnt aGϕ(an)=(ϕ(a)n)
ϕ(x)=ϕ((x)2)=(ϕ(x))2>0
Let x>0 then obtain x make sense
ϕ(x)=ϕ((x)2)=(ϕ(x))2<0
Thus, the funtion phi maps positive numers to positive numbers and negative to negative numbers.

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