1. The problem statement, all variables and given/known data
Let ##G## be a cyclic group of order ##7,## that is, ##G## consists of all ##a^i##, where ##a^7 = e.## Why is the mapping ##\phi:a^i\to a^{2i}## an automorphism of ##G## of order ##3##?
The attempt at a solution
I know the group ##G## is formed by the elements ##\{ e, a, a^2, a^3, a^4, a^5, a^6 \}##. Now under the mapping of ##\phi##, the order of elements is changed to: ##e, a^2, a^4, a^6, a, a^3, a^5 ##. Thus, ##\phi## is both one-one and onto. What I don't understand is why ##G## is of order ##3## by the map ##\phi##?
Let ##G## be a cyclic group of order ##7,## that is, ##G## consists of all ##a^i##, where ##a^7 = e.## Why is the mapping ##\phi:a^i\to a^{2i}## an automorphism of ##G## of order ##3##?
The attempt at a solution
I know the group ##G## is formed by the elements ##\{ e, a, a^2, a^3, a^4, a^5, a^6 \}##. Now under the mapping of ##\phi##, the order of elements is changed to: ##e, a^2, a^4, a^6, a, a^3, a^5 ##. Thus, ##\phi## is both one-one and onto. What I don't understand is why ##G## is of order ##3## by the map ##\phi##?
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