Let a b c and x be elements of a group G In each of the foll

Let a, b, c, and x be elements of a group G. In each of the following, solve for x in terms of a, b, and c.

Solution

Let a,b,c, and x be elements of a group G

(xax)³ =bx and x²a = (xa)^-1

(xax)³ =bx

((xax)³)^-2 =(bx)^-2

xax = x^-2 b^-2

Multiply left by x

x(xax) = x(x^-2 b^-2)

(xxa)x =( xx^-2 ) b^-2

(a¹x¹)x = (x¹)b² from  x²a = (xa)^-1 = a^-1 x^-1    and xx^-2 = (xx^-1)x^-1

a^₁(x¹x)=(x¹)b²

a^₁ = (x¹)b²

xa¹=b²

x= b^-2 a