Question

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

Solve simultaneously x^2 a=bxc^-1 and acx=xac.

Answer 1

We have

`x^2a=bxc^(-1)\implies x^2ac=bxc^(-1)c=bx`

`x^2ac=x(xac)=x(acx)=bx\implies xacxx^(-1)=bxx^(-1)\implies xac=b`

`x(ac)(ac)^(-1)=b(ac)^(-1)\implies x=b(ac)^(-1)\implies x=bc^(-1)a^(-1)`