- #1

Dromepalin

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**Group Theory: Prove o(b)|2 if ab = b^-1a**

## Homework Statement

Suppose G is a group and a, b [tex]\in[/tex] G

a) If o(a) is odd and a*b = b^−1*a, prove that o(b)|2.

b) If o(a) is even and a*b = b^−1*a, does it follow that o(b)|2? Prove your answer.

## Homework Equations

n/a

## The Attempt at a Solution

a) since ab = b^-1a, bab = a

(bab)^o(a) = a^o(a)

so, b^o(a)a^o(a)b^o(a) = e

[[I now realize I can't do this since G is not necessarily abelian, so not commutative..but ploughing on...]]

b^2o(a) = e, thus o(b) is even

I don't know how to approach this one. Been stuck on it for a frustrating amount of time now.

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