# Proving that every element of a monoid occurs exactly once

let $(B,\star)$ defines a monoid with a finite number of elements Let. the elements of $B$ be $\{x_1,x_2,x_3,x_4,\cdots\}$ where every element of $B$ occurs exactly once in this list

let $y$ be the invertible element of the monoid.

Prove that every element of the monoid occurs exactly once in this list $\{ y\star x_1,y \star x_2, \cdots, y\star x_n \}$.

Can anyone please point me in the right direction where to start? without telling me the answer.

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Start by trying to clearly and precisely state the problem. – Hurkyl Dec 27 '12 at 2:15
What do you mean it 'occurs exactly once'? – Alexander Gruber Dec 27 '12 at 2:17
@AlexanderGruber I have edited the question.Can you please advise me. – Jack welch Dec 27 '12 at 2:22
Hint: suppose that $y*x_i = y*x_j$, and show that $i$ must equal $j$. So the map $x_k \mapsto y*x_k$ is injective. – Josh Keneda Dec 27 '12 at 2:25

If $yx_i=yx_j$, what can you determine about $x_i$ and $x_j$, knowing that $y$ is invertible?