Demonstrate that each automorphism of&nbsp;ℝx, the group of nonzero real numbers under multiplication, translates from negative to negative and from positive to positive numbers.

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Demonstrate that each automorphism of&amp;nbsp;ℝx, the group of nonzero real numbers under multiplication, translates from negative to negative and from positive to positive numbers.

liannemdh · Accepted Answer

ϕ:R⋅→R⋅ 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 x∈R⋅ Let x&amp;gt;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 a∈Gϕ(an)=(ϕ(a)n) ϕ(x)=ϕ((x)2)=(ϕ(x))2&amp;gt;0 Let x&amp;gt;0 then obtain −x make sense ϕ(x)=ϕ((−x)2)=(ϕ(−x))2&amp;lt;0 Thus, the funtion phi maps positive numers to positive numbers and negative to negative numbers.

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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