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

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