This is from Abstract Algebra Let x be an element of odd fin

This is from Abstract Algebra:

Let x be an element of odd finite order in a group G. Show that x^2 has the same order as x.

Solution

Let G be a group and x G. We say g has finite order if xⁿ = e for some positive integer n.

The least n 1 such that xⁿ = e is called the order of x. If there is no such n (that is, xⁿ is not equal to zero for every n 1), we say x has infinite order.

xⁿ = e then order of g is n;

(xⁿ )² = e;

(xⁿ )*(xⁿ )=e

(xⁿ )=e (xⁿ )^-1 implies (xⁿ )= (xⁿ )-1

order of (xⁿ ) and inverse also same order.

(xⁿ )²   is also same order.