Question

If G is an infinite additive cyclic group with generator a. Prove that the equation x+x =a has no solution in G.

Short answer

Answer

The equationx+x =a has no solution in G

Step-by-step solution

Solution Step 1: Proving that the equation x+x =a has no solution in G.

$SupposeG=〈a〉fora\in G,wherea\ne 0,becauseifa=0,GisfinitebutitisgiventhatGisinfinite.$

Let’s assume that x + x =a has solution in G……….. (1)$Takingx\in G,suchthatx+x=a$

Because G is cyclic, there must be a positive integer n such that,

$x=a+a+a+a\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.....uptontimes..(2)}$

or,

$x=\left(-a\right)+\left(-a\right)+\left(-a\right)+\left(-a\right).....uptontimes..\left(3\right)$

Putting the value of x in (1) from (2),

$\begin{array}{l}\underset{upto2ntimes}{a+a+aa}=a\\ \underset{upto2n-1times}{a+a+aa}=0\\ Now,puttingthevaluexin\left(1\right)from\left(3\right),\end{array}$

$\begin{array}{l}\underset{upto2ntimes}{\left(-a\right)+\left(-a\right)+\left(-a\right)}=a\\ \underset{upto2n+1times}{\left(-a\right)+\left(-a\right)+\left(-a\right)}=a\end{array}$

Since the results arrived in the two conditions are contradictory, so G is finite. But, as G is given infinite, we can say that the equation x+x=a has no solution in G.