Proving $(G,*)$ is a Group: Hints and Tips for a Simple Group Theory Problem

[alexmahone]

Let $G$ be a set and $*$ a binary operation on $G$ that satisfies the following properties:

(a) $*$ is associative,

(b) There is an element $e\in G$ such that $e*a=a$ for all $a\in G$,

(c) For every $a\in G$, there is some $b\in G$ such that $b*a=e$.

Prove that $(G, *)$ is a group.

My attempt: I know we're supposed to show that $a*e=a$ for (b) and that $a*b=e$ for (c). But I can't figure out how.

Any suggestions? Hints only as this is an assignment problem.

 

[caffeinemachine]

Let $G$ be a set and $*$ a binary operation on $G$ that satisfies the following properties:

(a) $*$ is associative,

(b) There is an element $e\in G$ such that $e*a=a$ for all $a\in G$,

(c) For every $a\in G$, there is some $b\in G$ such that $b*a=e$.

Prove that $(G, *)$ is a group.

My attempt: I know we're supposed to show that $a*e=a$ for (b) and that $a*b=e$ for (c). But I can't figure out how.

Any suggestions? Hints only as this is an assignment problem.

Hint for part (a). Pick $a\in G$ and let $a*e=b$. We want to show that $b=a$.

We know that there is a $c\in G$ such that $c*a=e$. Thus $c*a*e=c*b$, which gives $e=c*b$. We also have $e=c*a$. Now can you see what to do?

Try part (b) with this approach.

 

[alexmahone]

Got it, thanks!