Give an example of a group G and elements a , b ∈ G such that a^{-1}(ba) ≠ b.

Question

Give an example of a group G and elements $a,b ∈ G$ such that $a^{-1}ba \not=b$.

Any ideas as to how I would go about finding it?

Thanks

Answer 1

$a^{-1}ba \not = b$ is the same as $ab \not = ba$, which is just saying that the group is not abelian. So you can just look at any non-abelian group and try some non-identity elements.

Answer 2

E.g. let $G$ be the set of all invertible real $2 \times 2$ matrices and think about diagonalization.