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.

Answer 1

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.$x^2a=bxc^{-1}$

$x^2ac=bxc^{-1}c=bx$

$x^{-1}x^2ac=x^{-1}bx=b$ ($x^{-1}bx=b$)

$xac=b$

$xacc^{-1}=bc^{-1}$

$xa=bc^{-1}$    ($cc^{-1}=$)

$xaa^{-1}=bc^{-1}a^{-1}$

$x=bc^{-1}a^{-1}$

b.$acx=xac$

$acxc^{-1}=xacc^{-1}=xa$  ($cc^{-1}=1,cxc^{-1}=x$)

$axa^{-1}=xaa^{-1}$  ($aa^{-1}=1,axa^{-1}=x$)

$x=x$

Identity equation

Hence, given equation has infinite solution and satisfied for all values of a and c.