Suppose that 〈G ,*〉 is a group. Prove or give a counter example:
(∀a ,b,c∈G)( ( c∗a=c∗b)⇒(a=b) )

Question

Suppose that 〈G ,*〉 is a group. Prove or give a counter example:
(∀a ,b,c∈G)( ( c∗a=c∗b)⇒(a=b) )

anonymous · Answer

@Hunus

anonymous · Answer

hint, multiply both sides by $c^{-1}$

anonymous · Answer

clear or no?

anonymous · Answer

I have no idea how to do this. So I have no idea what multiplying c^-1 does to the problem @satellite73

anonymous · Answer

it gives you the equality you want

terenzreignz · Answer

We have what are called group axioms, one of them being, if <G, *> is a group, then 
$$\large \forall x \in G, \quad \exists x^{-1} \in G $$
such that
$$\large x*x^{-1} = e$$
where e is the identity element in G.

anonymous · Answer

$$c\circ a =c \circ b\implies c^{-1}\circ c\circ a=c^{-1}\circ c\circ b $$
$$\implies e\circ a=e\circ b\implies a=b$$

anonymous · Answer

Okay. I understand. Is this sufficient for a proof? @satellite73 @terenzreignz

terenzreignz · Answer

Yeah, but I reckon this was just a sketch... your actual proof has to be a teensy bit more formal...

anonymous · Answer

what else do you need to say? i even wrote the $e$ part which is not really necessary

terenzreignz · Answer

Details... like, since $c$ is an element of a group G, then $c^{-1}$ exists etc... you know, for nitpicky instructors...

anonymous · Answer

aah yes, i see

anonymous · Answer

be a hell of a group if you couldn't solve though, wouldn't it ?

anonymous · Answer

Thank you @satellite73 @terenzreignz

terenzreignz · Answer

:)
my experience with groups is minimal... but they're awesome :D

anonymous · Answer

:) haha thanks again @terenzreignz