Theoretical Algebra Proof o(g) = o (g^-1)?
I am trying to prove that the order of g is equal to the order of the inverse of g. The Given statement reads as such:
Prove that o(g) = o(g^-1) for any element g of a group G."
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Since gg^(-1) = g^(-1)g=e, we have that g^k (g^(-1))^k = e.
If the k = o(g^(-1)) < o(g), then g^k * e = g^k = e. This contradicts the fact that o(g) is the smallest positive power exponent of g giving the identity. Hence, they are equal.
I hope that helps!