prove that the products ab and ba are conjugate elements in a group?

Question by Jenny H: prove that the products ab and ba are conjugate elements in a group?

prove that the products ab and ba are conjugate elements in a group. Please show steps!

Best answer:

Answer by Copestone
The answer is almost tirivial. It is just a two line proof but since you ask for details, I would write down everything for you, including definitions.

Let a, b ? G, a group. Let e be the identity element.

Recall from definition that the products ab and ba are conjugates if there is an element x ? G such that
x(ab) = (ba)x

So, simply take x = a^{-1}

Then x(ab) = (a^{-1})(ab) = (a^{-1})(a) = eb = b
while (ba)x = (ba)(a^{-1}) = b((a)(a^{-1}) = be = b.

It follows that x(ab) = (ba)x.
Equivalently, some textbook also writes x(ab)x^{-1} = ba.

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