Cancellation and Neutral Elements in Groups: Solving Equations

Easy

Let $G$ be a group with composition law $\circ$.

The property $a\circ x=x\circ a=x$ for all $a\in G$

Select Optionis sometimes satisfied by more than one $x\in G$is always satisfied by only one $x\in G$is sometimes not satisfied by any $x\in G$

. For $a\in G$, the equation $a\circ x=a\circ y$

Select Optionsometimes has a solution $x\not=y$ in $G$always has only the solution $x=y$ in $G$sometimes doesn't hold for any $x,y$ in $G$

. For all $a\in G$, the equation $x\circ a=y\circ a$

Select Optionsometimes has a solution $x\not=y$ in $G$always has only the solution $x=y$ in $G$sometimes doesn't hold for any $x,y$ in $G$

. For all $a\in G$, the equation $x\circ a\circ u=y\circ a\circ v$

Select Optionsometimes has a solution with $x\not=y$, $u\not=v$ in $G$always has only the solution $x=y$, $u=v$ in $G$sometimes has no solution for any $x,y,u,v$ in $G$