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

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.