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, and c.

Answer 1

Answer:

$x=c.a^{-1}.b^{-1}$

Step-by-step explanation:

Given equation,

$axb = c-----(1)$

$(axb) ( a^{-1}.b^{-1} ) = c.(a^{-1}b{-1})$

$(xb)(a)(a^{-1}.b^{-1} ) = c.(a^{-1}b{-1})$ ( commutative law of multiplication )

$(xb)(a.a^{-1}).b^{-1} = c.a^{-1}b{-1}$ ( Associative property )

$(xb).e.b^{-1} = c.a^{-1}b{-1}$

$(xb).b^{-1}=c.a^{-1}.b^{-1}$   ( identity property )

$x.(b.b^{-1})=c.a^{-1}.b^{-1}$  ( Associative property )

$x.e=c.a^{-1}.b^{-1}$      

$x=c.a^{-1}.b^{-1}$                 ( identity property )