# Thread: Solvability, cancellation, identity, and inverse properties

1. ## Solvability, cancellation, identity, and inverse properties

Prove that if G is a set with an associative operation which has the solvability and cancellation properties, then it has the identity and inverse properties.

Solvability: for any elements $\displaystyle a$, $\displaystyle b$ of $\displaystyle G$, there is an element $\displaystyle c$ in $\displaystyle G$ with $\displaystyle a * c = b$, and an element $\displaystyle d$ in $\displaystyle G$ with $\displaystyle d * a = b$
Cancellation: for any three elements $\displaystyle a, b, c$ of $\displaystyle G$, if $\displaystyle a * b = a * c$, then $\displaystyle b = c$
Existence of Identity: there is an element $\displaystyle e$ of $\displaystyle G$ so that for all $\displaystyle a$ in $\displaystyle G$, $\displaystyle e * a = a * e = a$
Existence of Inverse: for any $\displaystyle a$ in $\displaystyle G$, there is an element $\displaystyle b$ in $\displaystyle G$ so that $\displaystyle a * b = b * a = e$

2. Originally Posted by Zennie
Prove that if G is a set with an associative operation which has the solvability and cancellation properties, then it has the identity and inverse properties.

Solvability: for any elements $\displaystyle a$, $\displaystyle b$ of $\displaystyle G$, there is an element $\displaystyle c$ in $\displaystyle G$ with $\displaystyle a * c = b$, and an element $\displaystyle d$ in $\displaystyle G$ with $\displaystyle d * a = b$
Cancellation: for any three elements $\displaystyle a, b, c$ of $\displaystyle G$, if $\displaystyle a * b = a * c$, then $\displaystyle b = c$
Existence of Identity: there is an element $\displaystyle e$ of $\displaystyle G$ so that for all $\displaystyle a$ in $\displaystyle G$, $\displaystyle e * a = a * e = a$
Existence of Inverse: for any $\displaystyle a$ in $\displaystyle G$, there is an element $\displaystyle b$ in $\displaystyle G$ so that $\displaystyle a * b = b * a = e$
What have you tried? What about solving the equation $\displaystyle dc=c$?