Answer with Step-by-step explanation:
We are given that a, b, c and x are elements in the group G.
We have to find the value of x in terms of a, b and c.
a.[tex]x^2a=bxc^{-1}[/tex]
[tex]x^2ac=bxc^{-1}c=bx[/tex]
[tex]x^{-1}x^2ac=x^{-1}bx=b[/tex] ([tex]x^{-1}bx=b[/tex])
[tex]xac=b[/tex]
[tex]xacc^{-1}=bc^{-1}[/tex]
[tex]xa=bc^{-1}[/tex] ([tex]cc^{-1}=[/tex])
[tex]xaa^{-1}=bc^{-1}a^{-1}[/tex]
[tex]x=bc^{-1}a^{-1}[/tex]
b.[tex]acx=xac[/tex]
[tex]acxc^{-1}=xacc^{-1}=xa[/tex] ([tex]cc^{-1}=1,cxc^{-1}=x[/tex])
[tex]axa^{-1}=xaa^{-1}[/tex] ([tex]aa^{-1}=1,axa^{-1}=x[/tex])
[tex]x=x[/tex]
Identity equation
Hence, given equation has infinite solution and satisfied for all values of a and c.
