let x be an element of group g. Prove that if |x|=n then x^-1=x^n-1

Let x, y be elements in a group G. Prove thatx^(−1). y^n. x = (x^(−1).yx)^nfor all n ∈ Z.

Consider the group G={1,−1,i,−i} under multiplication, where i^2=−1. Determine the inverse of each element in the group.

Let G be the cyclic group Z_4 under addition, of integers modulo 4. Let H be the subgroup of multiples of 2, {0,2}. Write out the two distinct cosets of H. One is 0+H=H. The other is some g+H. Then write out the addition table for the quotient group G/H consisting of those two cosets. The sum of a+H and b+H will be a+b+H.