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Homework Statement
Prove that in any group the orders of ab and ba equal.
Homework Equations
n/a
The Attempt at a Solution
Let (ab)^{x} = 1.
Using associativity, we get
(ab)^{x} = a(ba)^{x-1}b = 1.
Because of the existence of inverses--namely a^{-1} and b^{-1}--this implies
(ba)^{x-1} = a^{-1}b^{-1} = (ba)^{-1}.
Multiplying both sides by (ba) = ((ba)^{-1})^{-1} yields
(ba)^{x} = 1.
So,
(ab)^{x} = (ba)^{x} = 1,
and the orders ab and ba are the same.
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How is that?
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