If in a finite group every element is conjugate to its inverse, then show that all the entries in the character table are real.

Let G be a group and a ∈ G. A conjugate of a is any element of the form bab-1 for some b ∈ G. Show that if c = bab-1 is a conjugate of a in G and n is any integer, then cn=banb-1.  First prove for n ≥ 0 by induction on n.

In an abstract algebra equation about groups, is "taking the inverse of both sides of an equation" an acceptable operation?  I know you can right/left multiply equations by elements of the group, but was wondering if one can just take the inverse of both sides?

For any field, the nonzero elements form a group under the field multiplication? T or F